Research Article Existence Results for the Periodic Thomas ...downloads.hindawi.com › journals › amp › 2015 › 652407.pdf · Research Article Existence Results for the Periodic

Research ArticleExistence Results for the Periodic Thomas-Fermi-Dirac-vonWeizsaumlcker Equations

Shaowei Chen1 Lishan Lin2 and Liqin Xiao1

1School of Mathematical Sciences Huaqiao University Quanzhou 362021 China2School of Applied Mathematical Sciences Xiamen University of Technology Xiamen 361024 China

Correspondence should be addressed to Shaowei Chen chenswamssaccn

Received 18 October 2014 Accepted 23 December 2014

Academic Editor Hagen Neidhardt

Copyright copy 2015 Shaowei Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider the Thomas-Fermi-Dirac-von Weizsacker equation minusΔ119906 + 119881(119909)119906 + (1199062 ⋆ (1|119909|))119906 = 120582|119906|119901minus2119906 minus |119906|119902minus2119906 119906 isin 1198671(R3)where 120582 gt 0 is a parameter 2 lt 119901 lt 119902 lt 6 119881 isin 119871infin(R3) is 1-periodic in 119909

119895 for 119895 = 1 2 3 and 0 is in a spectral gap of the operator

minusΔ+119881 Using a new infinite-dimensional linking theorem we prove that for sufficiently small 120582 gt 0 this equation has a nontrivialsolution

1 Introduction and Statement of Results

In this paper we consider the following equation

minus Δ119906 + 119881 (119909) 119906 + (119906

2⋆

1

|119909|

) 119906

= 120582 |119906|

119901minus2119906 minus |119906|

119902minus2119906 119906 isin 119867

1(R3)

(1)

where 2 lt 119901 lt 119902 lt 6 120582 gt 0 is a parameter

119906

2⋆

1

|119909|

= int

R3

119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119910(2)

and 1198671(R3) is the standard Sobolev space with norm119906

1198671(R3) = (intR3

(|nabla119906|

2+ 119906

2)119889119909)

12The potential function 119881 isin 119871infin(R119873) is 1-periodic in 119909

119895

for 119895 = 1 2 3 Under this assumption 120590(119871) the spectrum ofthe operator

119871 = minusΔ + 119881 119871

2(R3) 997888rarr 119871

2(R3) (3)

is a purely continuous spectrum that is bounded below andconsists of closed disjoint intervals ([1 Theorem XIII100])Thus the complement R 120590(119871) consists of open intervalscalled spectral gaps We assume the following

(v) 119881 isin 119871infin(R119873) is 1-periodic in 119909119895for 119895 = 1 2 3 and 0 is

in a spectral gap (minus120572 120573) of minusΔ + 119881 where 0 lt 120572 120573 lt+infin

A solution 119906 of (1) is called nontrivial if 119906 equiv 0 Our mainresult is as follows

Theorem 1 Suppose that 2 lt 119901 lt 119902 lt 6 and (k) is satisfiedThen there exists 120582

0gt 0 such that for any 0 lt 120582 lt 120582

0 the

problem (1) has a nontrivial solution

Equation (1) arises in the study of the Thomas-Fermi-Dirac-von Weizsacker (TFDW) model for atoms andmolecules with no external potential where 120588 = 1199062 isthe electron density In [2 section VIII] Lieb studied theexistence and symmetric and asymptotic properties ofsolutions to

minus 119860Δ120595 + (119881 + 120572)120595 + (120595

2⋆ |119909|

minus1) 120595 + 120574120595

2119901minus1minus 119862

119890120595

53

= minus120583120595 in R3

(4)

for various choices of119881 Here119860 120572 120574 119862119890 and 120583 are constants

and 119901 gt 43 The term minus119862119890120595

53 is called the Dirac term

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 652407 9 pageshttpdxdoiorg1011552015652407

2 Advances in Mathematical Physics

with strength 119862119890 In [3] Le Bris considered the minimizing

problem

inf120595isin1198671(R3)int

R31205952119889119909=120582

119864 (120595) (5)

with

119864 (120595) = int

R3

1003816

1003816

1003816

1003816

nabla120595

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881 (119909) 120595

2119889119909

+ 119888

1int

R3120595

103119889119909 minus 119888

2int

R3120595

83119889119909

+ int

R3int

R3

120595

2(119909) 120595

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909

(6)

where 1198881and 119888

2are positive constant and 119881 is the Coulomb

potential created by the atomic nuclei Le Bris proved thatthere exists a 120582

119888gt 0 such that for 0 lt 120582 lt 120582

119888 the problem

admits a minimizer Moreover if 1198882is small enough then the

minimizer is unique Using the Lagrange multiplier it is easyto see that the minimizer of (5) is a solution to

minus Δ120595 + 119881 (119909) 120595 + (120595

2⋆ |119909|

minus1) 120595 + 119888

1120595

73minus 119888

2120595

53

= 120583120595 inR3(7)

for some 120583 However in a recent paper [4] by Lu and Ottothe authors proved that for sufficiently large 120582 the variationalproblem (5) with 119881 = 0 does not have a minimizer Equation(1) with 119881 a periodic function is used to describe a Hartreemodel for crystals (see [5 6]) Moreover when 120582 = 0 (1) isoften referred to theThomas-Fermi-vonWeizsackermodel inthe literature One can see [7] and references therein Finallywe should mention that the so-called Schrodinger-Poisson-Slater equation

minusΔ119906 + 119881 (119909) 119906 + (119906

2⋆

1

|119909|

) 119906 = 120582 |119906|

119901minus2119906 in R

3 (8)

is also related to (1) and has attractedmuch attention in recentyears (see [5 7ndash22])

The variational functional for (1) is given by

119869

120582(119906) =

1

2

int

R3(|nabla119906|

2+ 119881 (119909) 119906

2) 119889119909

+

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus

120582

119901

int

R3|119906|

119901119889119909 +

1

119902

int

R3|119906|

119902119889119909 119906 isin 119867

1(R3)

(9)

In other words the critical points of 119869120582are solutions to (1)

Hence it is natural to use critical point theory to obtainsolutions to (1) Under assumption (v) the quadratic formint

R3(|nabla119906|

2+ 119881119906

2)119889119909 has infinite-dimensional positive and

negative spaces It can be shown that for sufficiently small120582 gt 0minus119869

120582has a global infinite-dimensional linking geometry

(for its definition see (64) of Willem [23]) However thedifficulty rises when the classical infinite-dimensional linkingtheorem (see [23]) is used to obtain a critical point to 119869

120582

This theorem requires the functional to satisfy some uppersemicontinuous assumption (see (63) of [23]) Howeverbecause the nonlinearity

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 +

1

119902

int

R3|119906|

119902119889119909 minus

120582

119901

int

R3|119906|

119901119889119909

(10)

is neither positive definite nor negative definite in 1198671(R3)whenever 120582 gt 0 minus119869

120582does not satisfy the upper semicontin-

uous assumption To overcome this difficulty we use a newinfinite-dimensional linking theorem in [24] to obtain a (119862)

119888

sequence (see Definition 6) for minus119869120582 We can prove that a (119862)

119888

sequence for minus119869120582is bounded in1198671(R3) if 120582 gt 0 is sufficiently

small This result is new and original Finally through theconcentration-compactness principle and the (119862)

119888sequence

a nontrivial solution to (1) is obtained Our method can beused to study more general equation like

minusΔ119906 + 119881 (119909) 119906 + (119906

2⋆

1

|119909|

) 119906 = 119891 (119909 119906) 119906 isin 119867

1(R3)

(11)

Notation 119861119903(119886) denotes an open ball of radius 119903 and center

119886 For a Banach space 119883 we denote the dual space of 119883 by119883

1015840 and denote strong and weak convergence in119883 by rarr and respectively For 120593 isin 1198621(119883R) we denote the Frechetderivative of 120593 at 119906 by 1205931015840(119906) The Gateaux derivative of 120593 isdenoted by ⟨1205931015840(119906) V⟩ for all 119906 V isin 119883 119871119902(R3) and 119871119902loc(R

3)

denote the standard 119871119902 space and the locally 119902-integrablefunction space respectively (1 le 119902 le infin) Let Ω be a domaininR119873 (119873 ge 1) 119862infin

0(Ω) is the space of infinitely differentiable

functions with compact support in Ω We use 119874(ℎ) and 119900(ℎ)to mean |119874(ℎ)| le 119862|ℎ| and 119900(ℎ)|ℎ| rarr 0 respectively

2 Variational Setting for (1)

LetD12(R3) be the Hilbert space

D12(R3) = 119906 isin 119871

6(R3) | int

R3|nabla119906|

2119889119909 lt infin (12)

with inner product

(119906 V)D12(R3) = intR3nabla119906nablaV 119889119909 (13)

For 119906 isin 1198671(R3) by the Lax-Milgram theorem the equation

minusΔ120601 = 4120587119906

2 in R3 (14)

has a unique solution 120601119906isin D12(R3) (see Proposition 22 of

[10]) And byTheorem 221 of [25] 120601119906can be expressed by

120601

119906(119909) = (119906

2⋆

1

1003816

1003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

) (119909) = int

R3

119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119910 119909 isin R3

(15)

The function 120601119906has the following properties

Advances in Mathematical Physics 3

Lemma 2 (i) There exists a positive constant 119862 such that forany 119906 isin 1198671(R3)

1003817

1003817

1003817

1003817

120601

119906

1003817

1003817

1003817

1003817D12(R3)le 119862 119906

2

1198671(R3) (16)

int

R3

1003816

1003816

1003816

1003816

nabla120601

119906

1003816

1003816

1003816

1003816

2

119889119909 = 4120587int

R3120601

119906119906

2119889119909

= 4120587int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862 119906

4

119871125(R3)le 119862 119906

4

1198671(R3)

(17)

(ii) For any 119906 isin 1198671(R3) 120601119906ge 0 in R3

Proof The inequality

4120587int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 le 119862 119906

4

119871125(R3)

(18)

can be derived from the the Hardy-Littlewood-Sobolevinequality (see (11) of [26]) by choosing 119899 = 3 120582 = 1 119901 =119905 = 65 and 119891 = 119892 = 1199062 there For a proof of the otherproperties of 120601

119906in this lemma one can see [8 10 12]

By this lemma the functional 119869120582in (9) is well defined in

119867

1(R3) And a direct computation shows that the derivative

of 119869120582is

⟨119869

1015840

120582(119906) V⟩ = int

R3(nabla119906nablaV + 119881 (119909) 119906V) 119889119909

+ int

R3int

R3

119906 (119909) V (119909) 1199062 (119910)1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus 120582int

R3|119906|

119901minus2119906V 119889119909 + int

R3|119906|

119902minus2119906V 119889119909

forall119906 V isin 1198671 (R3)

(19)

It is easy to verify that 119869120582is a 1198621 functional in 1198671(R3)

Moreover we have the following

Lemma 3 The following statements are equivalent

(i) 119906 isin 1198671(R3) is a solution of (1)(ii) 119906 is a critical point of 119869

120582

Under the assumption (k) there is a standard variationalsetting for the quadratic formint

R3(|nabla119906|

2+119881(119909)119906

2)119889119909 One can

see section 64 of [23] But for the convenience of the readerwe state it here

Let 119871 be the operator defined by (3) We denote by |119871|12

the square root of the absolute value of119871The domain of |119871|12is the space

119883 = 119867

1(R3) (20)

On 119883 we choose the inner product (119906 V) = intR3|119871|

12119906 sdot

|119871|

12V 119889119909 and the corresponding norm 119906 = radic(119906 119906) Since

0 lies in a gap of the essential spectrum of 119871 there exists anorthogonal decomposition 119883 = 119884 oplus 119885 such that 119884 and 119885 arethe positive and negative spaces corresponding to the spectraldecomposition of 119871 Since 119881 is 1-periodic for all variablesthey are invariant under the action ofZ3 that is for any 119906 isin 119884or 119906 isin 119885 and for any k = (119899

1 119899

2 119899

3) isin Z3 119906(sdot minus k) is also in

119884 or 119885 Furthermore

forall119906 isin 119884 int

R3(|nabla119906|

2+ 119881119906

2) 119889119909 = (119906 119906) = 119906

2


R3(|nabla119906|

2+ 119881119906

2) 119889119909 = minus (119906 119906) = minus 119906

2

(21)

Let 119876 119883 rarr 119885 119875 119883 rarr 119884 be the orthogonal projectionsBy (21)

int

R3(|nabla119906|

2+ 119881119906

2) 119889119909 = 119875119906

2minus 119876119906

2 forall119906 isin 119883 (22)

Moreover by119883 = 119884 oplus 119885 we have

119906 = 119875119906 + 119876119906 119906

2= 119875119906

2+ 119876119906

2 forall119906 isin 119883

(23)

By (22) and (9)

minus119869

120582(119906) =

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+

120582

119901

int

R3|119906|

119901119889119909 minus

1

119902

int

R3|119906|

119902119889119909 119906 isin 119883

(24)

Moreover by (19)

⟨minus119869

1015840

120582(119906) V⟩ = (119876119906 V) minus (119875119906 V)

minus int

R3int

R3

119906 (119909) V (119909) 1199062 (119910)1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ 120582int

R3|119906|

119901minus2119906V 119889119909

minus int

R3|119906|

119902minus2119906V 119889119909 forall119906 V isin 119883

(25)

3 A Global Linking Geometry for minus119869120582

Let 119890119896 be a total orthonormal sequence in 119884 and

|119906| = max119876119906 infin

sum

119896=1

1

2

119896+1

1003816

1003816

1003816

1003816

(119875119906 119890

119896)

1003816

1003816

1003816

1003816

(26)

For 119877 gt 119903 gt 0 and 1199060isin 119885 with 119906

0 = 1 set

119873 = 119906 isin 119885 | 119906 = 119903

119872 = 119906 + 119905119906

0| 119906 isin 119884 119905 ge 0

1003817

1003817

1003817

1003817

119906 + 119905119906

0

1003817

1003817

1003817

1003817

le 119877

120597119872 = 119906 isin 119884 | 119906 le 119877

cup 119906 + 119905119906

0| 119906 isin 119884 119905 gt 0

1003817

1003817

1003817

1003817

119906 + 119905119906

0

1003817

1003817

1003817

1003817

= 119877

(27)


Definition 4 Let 120601 isin 1198621(119883R) 1206011015840 is called weakly sequen-tially continuous if 119906 isin 119883 and 119906

119899 sub 119883 are such that 119906

119899 119906

then for any 120593 isin 119883 ⟨1206011015840(119906119899) 120593⟩ rarr ⟨120601

1015840(119906) 120593⟩

Lemma 5 The functional minus119869120582satisfies the following

(a) minus1198691015840120582is weakly sequentially continuous and for every

120582 gt 0

sup119872

(minus119869

120582) lt +infin (28)

(b) There exist 120575 gt 0 119877 gt 119903 gt 0 1199060isin 119885 with 119906

0 = 1

and 12058210158400gt 0 such that for 0 lt 120582 lt 1205821015840

0

inf119873

(minus119869

120582) gt maxsup

120597119872

(minus119869

120582) sup|119906|le120575

(minus119869

120582(119906)) (29)

Proof (a) Let 119906 isin 119883 and 119906119899 sub 119883 be such that 119906

119899 119906 as

119899 rarr infin It follows that

(119876119906

119899 V) 997888rarr (119876119906 V) (119875119906

119899 V) 997888rarr (119875119906 V)

119899 997888rarr infin forallV isin 119883(30)

Since 119906119899 119906 implies 119906

119899rarr 119906 in 119871119904loc(R

3) for any 1 le 119904 lt

6 we get that for any V isin 119862infin0(R3) as 119899 rarr infin

int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899V 119889119909 997888rarr int

R3|119906|

119901minus2119906V 119889119909

int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899V 119889119909 997888rarr int

R3|119906|

119902minus2119906V 119889119909

(31)

By Lemma 2 we have that 120601119906119899 is bounded in D12(R3)

Therefore up to a subsequence 120601119906119899 120601

119906as 119899 rarr infin It

follows that 120601119906119899rarr 120601

119906in 119871119904loc(R

3) for any 1 le 119904 lt 6 This

yields that for any V isin 119862infin0(R3) as 119899 rarr infin

int

R3120601

119906119899119906

119899V 119889119909 997888rarr int

R3120601

119906119906V 119889119909 (32)

From (25) (30) (31) and (32) we deduce that as 119899 rarr infin

⟨minus119869

1015840

120582(119906

119899) V⟩ 997888rarr ⟨minus1198691015840

120582(119906) V⟩ forallV isin 119862infin

0(R3) (33)

Hence minus1198691015840120582is weakly sequentially continuous Moreover it is

easy to see that minus119869120582maps bounded sets into bounded sets

hence sup119872(minus119869

120582) lt +infin

(b) If 119906 isin 119885 then 119875119906 = 0 and 119876119906 = 119906 Using (17) we getfrom (24) that for any 119906 isin 119885

minus119869

120582(119906) ge

1

2

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus

1

119902

int

R3|119906|

119902119889119909

ge

1

2

119906

2minus

119862

16120587

119906

4minus

119862

1

119902

119906

119902

(34)

where 1198621comes from the Sobolev inequality 119906

119871119902(R3) le

119862

1119902

1119906 for all 119906 isin 119883 It follows that we can choose

sufficiently small 119903 gt 0 such that for119873 = 119906 isin 119885 | 119906 = 119903

inf119873

(minus119869

120582) gt

1

4

119903

2gt 0 (35)

Since 119902 gt 119901 there exists Λ gt 0 such that120582

119901

|119905|

119901minus

1

119902

|119905|

119902lt 0 if 0 lt 120582 le 1 |119905| gt Λ (36)

Let 1198621015840 gt 0 be such that

1199061198712(R3) le 119862

1015840119906 forall119906 isin 119883

(37)

From 119902 gt 119901 gt 2 we deduce that there exists 0 lt 12058210158400lt 1 such

that for any 0 lt 120582 le 12058210158400

sup|119905|leΛ

119905

minus2(

120582

119901

|119905|

119901minus

1

119902

|119905|

119902) lt

1

4119862

10158402 (38)

From (36) and (38) we deduce that for any 0 lt 120582 le 12058210158400and

119906 isin 119883minus 119869

120582(119906)

=

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ int

R3(

120582

119901

|119906|

119901minus

1

119902

|119906|

119902)119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ int

119909||119906(119909)|leΛ

(

120582

119901

|119906|

119901minus

1

119902

|119906|

119902)119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+

1

4119862

10158402int

R3119906

2119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2

minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 +

1

4

119906

2

=

3

4

119876119906

2minus

1

4

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

(39)

FromTheorem 11 of [27] we deduce that there exists 119862lowastgt 0

such that for any 119906 isin 119883

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

ge 119862

lowast(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

(40)


Let 1199060isin 119885 be such that 119906

0 = 1 And let 119906 = V + 119905119906

0isin

119884 oplusR1199060 Since there exists a continuous projection

119871

2(R3 |119909|

minus12(1 + |ln |119909||)minus1) 997888rarr R119906

0 (41)

we deduce that there exists a constant 119862lowastlowastgt 0 such that for

any 119906 = V + 1199051199060isin 119884 oplusR119906

0

(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

ge 119862

lowastlowast119905

4

(42)

Combining (39)ndash(42) we deduce that there exists a constant120581 gt 0 such that for any 119906 = 119905119906

0+ V isin 119884 oplusR119906

0

minus119869

120582(119906) le

3

4

119905

2minus

1

4

V2 minus 1205811199054 (43)

It follows that

minus119869

120582(119906) 997888rarr minusinfin as 119906 997888rarr infin 119906 isin 119884 oplusR1199060 (44)

Moreover for 0 lt 120582 le 12058210158400and 119906 isin 119884 (39) implies

minus119869

120582(119906) le minus

1

4

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 le 0

(45)

Together with (44) this implies that there exists 119877 gt 119903 suchthat

sup120597119872

(minus119869

120582) le 0 lt inf

119873

(minus119869

120582) (46)

From (39) and the definition of | sdot | (see (26)) we havefor 0 lt 120582 le 1205821015840

0

minus119869

120582(119906) le

3

4

119876119906

2le

3

4

|119906|

2

(47)

Choosing 120575 = 119903radic6 (47) and (35) give that

sup|119906|le120575

(minus119869

120582(119906)) lt inf

119873

(minus119869

120582) (48)

Together with (46) this yields (29)

4 Boundedness of (119862)119888

Sequence of minus119869120582

Definition 6 Let 120601 isin 1198621(119883R) A sequence 119906119899 sub 119883 is called

a (119862)119888sequence for 120601 if

sup119899

120601 (119906

119899) le 119888 (1 +

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

)

1003817

1003817

1003817

1003817

1003817

120601

1015840(119906

119899)

1003817

1003817

1003817

1003817

10038171198831015840997888rarr 0

as 119899 997888rarr infin(49)

In this section we will prove that if 120582 gt 0 is sufficientlysmall then a (119862)

119888sequence for minus119869

120582is bounded in119883

Since 119902 gt 119901 gt 2 there exists119863 gt 0 such that

|119905|

119902minus2minus 120582 |119905|

119901minus2gt

1003817

1003817

1003817

1003817

119881

minus

1003817

1003817

1003817

1003817119871infin(R3)

if 0 lt 120582 le 1 |119905| gt 1198632

(50)

where 119881minus(119909) = maxminus119881(119909) 0 119909 isin R3

Lemma 7 Suppose that 0 lt 120582 le 1 If 119906119899 is a (119862)

119888sequence

for minus119869120582 then

lim119899rarrinfin

int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (51)

Proof Let V119899= max119906

119899(119909) minus 1198632 0 It is easy to verify that

V119899isin 119883 and V

119899 le 119906

119899 for all 119899 isin N Together with the fact

that 119906119899 is a (119862)

119888sequence of minus119869

120582 this implies ⟨1198691015840(119906

119899) V119899⟩ =

119900(1) where 119900(1) denotes the infinitesimal depending only on119899 that is 119900(1) rarr 0 as 119899 rarr infin Choosing V = V

119899in (19)

by (50) and the fact that 120601119906119899 119906119899 and V

119899are nonnegative in

119909 | 119906

119899(119909) gt 1198632 we have

119900 (1) = ⟨119869

1015840(119906

119899) V

119899⟩

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881 (119909) 119906

119899V119899119889119909

+ int

R3120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

R3(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

) 119906

119899V119899119889119909

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881

+119906

119899V119899119889119909

+ int

119909|119906119899(119909)gt1198632

120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

119909|119906119899(119909)gt1198632

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

minus 119881

minus(119909) ) 119906

119899V119899119889119909

ge int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 ge

119862(int

R3

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

=

119862(int

119909|119906119899(119909)gt1198632

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

(52)

where 119881+= 119881 + 119881

minusge 0 in R3 and 119862 is the Sobolev constant

Note that V119899ge 119906

1198992 on 119909 | 119906

119899(119909) gt 119863 Together with (52)

this implies

lim119899rarrinfin

int

119909|119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (53)

Similarly we can prove

lim119899rarrinfin

int

119909|minus119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (54)

The result of this lemma follows from these two limits

Lemma 8 There exists 120582101584010158400gt 0 such that if 0 lt 120582 lt 12058210158401015840

0and

119906

119899 is a (119862)


120582 then 119906

119899 is bounded in119883

Proof By (1 + 119906119899) minus 119869

1015840

120582(119906

119899)

1198831015840 rarr 0 we have

⟨minus119869

1015840

120582(119906

119899) 119876119906

119899⟩ = 119900 (1) ⟨minus119869

1015840

120582(119906

119899) 119875119906

119899⟩ = 119900 (1)

(55)


Choosing V = 119876119906119899and V = 119875119906

119899in (25) (55) implies that

1003817

1003817

1003817

1003817

119876119906

119896

1003817

1003817

1003817

1003817

2

= int

R3119906

119899120601

119906119899sdot 119876119906

119899119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 + 119900 (1)

1003817

1003817

1003817

1003817

119875119906

119899

1003817

1003817

1003817

1003817

2

= minus int

R3119906

119899120601

119906119899sdot 119875119906

119899119889119909

+ int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 + 119900 (1)

(56)

By these two equalities and 119906119899

2= 119875119906

119899

2+ 119876119906

119899

2 (see(23)) we obtain

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

(57)

Let

120594

1198991(119909) =

1 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

le 119863

0 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

gt 119863

(58)

and 1205941198992= 1 minus 120594

1198991 where 119863 comes from (50) Then 1199062

119899=

(119906

119899120594

1198991)

2+ (119906

119899120594

1198992)

2 and

120601

119906119899= 120601

1199061198991205941198991+ 120601

1199061198991205941198992 (59)

Since 1206011199061198991205941198991

is a solution of the equation

minusΔ120601 = 4120587 (119906

119899120594

1198991)

2 in R3

(60)

by (119906119899120594

1198991)

2le 119863

2 in R3 and the standard elliptic estimates(see [28 Theorem 817]) we get that there exists a positiveconstant 119862

2that is independent of 119899 and 119910 isin R3 such that

for any 119910 isin R3

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(1198611(119910))

le 119862

2(int

1198612(119910)

1003816

1003816

1003816

1003816

119906

119899120594

1198991

1003816

1003816

1003816

1003816

4

119889119909)

12

le 119862

2119863

2(int

1198612(119910)

119889119909)

12

(61)

Let 1198623= (int

1198612(0)119889119909)

12 By (61)

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(R3)le 119862

2119862

3119863

2 (62)

Together with 1206011199061198991205941198991

ge 0 inR3 (see Lemma 2(ii)) this implies

0 le 120601

1199061198991205941198991le 119862

2119862

3119863

2 in R3 (63)

By (57) (59) and (63) we have

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

1199061198991205941198991sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|gt119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

le

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991119906

119899

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ (int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ 120582(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119901(R3)

+ (int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119902(R3)

(64)

Since 119876119906119899minus 119875119906

119899

1198712(R3) le 2119862

1015840119906

119899 (1198621015840 is the constant

coming from (37)) 119876119906119899minus 119875119906

119899

119871119901(R3) le 119862

10158401015840119906

119899 and

119876119906

119899minus 119875119906

119899

119871119902(R3) le 119862

10158401015840119906

119899 where 11986210158401015840 is a positive constant

independent of 119899 and 120582 we have1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840 10038171003817

1003817

1003817

1003817

119906

119899120601

1199061198991205941198991

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 120582119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(65)


By Lemma 7 we deduce that int119909||119906119899(119909)|gt119863

|119906

119899|

119901119889119909 = 119900(1) and

int

119909||119906119899(119909)|gt119863|119906

119899|

119902119889119909 = 119900(1) if 120582 le 1 Together with (63) this

implies that for 0 lt 120582 le 1

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(66)

Using the Hardy-Littlewood-Sobolev inequality (see (11) of[26]) we get that

int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

(119906

119899120594

1198992(119910))

2

sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

le 119862

101584010158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909)

56

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

(67)

where 119862119867119871119878

and 119862101584010158401015840 are positive constants independent of 119899and 120582 By Lemma 7 we have

int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909 = 119900 (1) if 0 lt 120582 le 1 (68)

Combining (68) and (67) with (66) yields that for 0 lt 120582 le 1

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(69)

Let 0 lt 120598 lt 119863 Then for 0 lt 120582 le 1

(int

119909||119906119899(119909)|le120598

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

le (int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus1

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus1

)

2

119889119909)

12

le 2(int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119901minus2

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119902minus2

) 119889119909)

12

le 4 (120598

119901minus2+ 120598

119902minus2) (int

119909||119906119899(119909)|le120598

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909)

12

le 4119862

1015840(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(70)

Combining (70) with (69) yields that for 0 lt 120582 le 1

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 4119862

10158402(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909|120598lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(71)

Let 1205980isin (0min1198631198631(119902minus119901) 1) be such that 411986210158402(120598119901minus2

0+

120598

119902minus2

0) lt 12 Then (71) implies that

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 4119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 4119862

1015840(int

119909|1205980lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(72)

From sup119899(minus119869

120582(119906

119899)) le 119888 and (1+119906

119899)minus119869

1015840

120582(119906

119899)

1198831015840 rarr 0

we obtain

119900 (1) + 119888 ge minus119869

120582(119906

119899) +

1

2

⟨119869

1015840

120582(119906

119899) 119906

119899⟩

=

1

4

int

R3119906

2

119899120601

119906119899119889119909

minus int

R3(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus (

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

(73)


It follows that

1

4

int

R3119906

2

119899120601

119906119899119889119909

+ int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + int

119909||119906119899(119909)|le1205821(119902minus119901)

(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus(

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

+ 119900 (1)

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

(74)

where 1198624is a positive constant depending only on 119901 and 119902

Because (12 minus 1119902)|119905|119902 minus120582(12 minus 1119901)|119905|119901 gt 0 if |119905| ge 1205821(119902minus119901)we get that for 0 lt 120582 le 120598119902minus119901

0

int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

(75)

Then by (74) and intR3119906

2

119899120601

119906119899119889119909 ge int

R3119906

2

119899120601

1199061198991205941198991119889119909 =

119906

119899120601

12

1199061198991205941198991

2

1198712(R3)

we get that for 0 lt 120582 le 120598119902minus1199010

1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(76)

It is easy to verify that there exists a positive constant 1198625

depending only on 119901 119902 119863 and 1205980such that if 0 lt 120582 le 120598119902minus119901

02

and 1205980le |119905| le 119863 then

119862

5(120582 |119905|

119901minus2119905 minus |119905|

119902minus2119905)

2

le (

1

2

minus

1

119902

) |119905|

119902minus 120582(

1

2

minus

1

119901

) |119905|

119901

(77)

Together with (76) this yields that for 0 lt 120582 le 120598119902minus11990102

1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ 119862

5int

119909|119863ge|119906119899(119909)|gt1205980

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(78)

Combining (78) with (72) we deduce that there exists 120582101584010158400isin

(0 120598

119902minus119901

02) that is small enough such that 119906

119899 is bounded if

0 lt 120582 lt 120582

10158401015840

0

5 Proof of Theorem 1

Proof of Theorem 1 Let 1205820= min1205821015840

0 120582

10158401015840

0 where 1205821015840

0and 12058210158401015840

0

are the constants in Lemmas 5 and 8 respectively By Lemmas5 and 8 andTheorem 13 of [24] we deduce that for any 0 lt120582 lt 120582

0 there exists a bounded (119862)

119888sequence 119906

119899 for minus119869

120582

with inf119899|119906

119899| gt 0 Up to a subsequence either

(a) lim119899rarrinfin

sup119910isinR3 int1198611(119910)

|119906

119899|

2119889119909 = 0 or

(b) there exist 984858 gt 0 and 119886119899isin Z3 such that

int

1198611(119886119899)|119906

119899|

2119889119909 ge 984858

If (a) occurs the Lions lemma (see [23 Theorem 121])implies that 119906

119899satisfies 119906

119899rarr 0 in 119871119904(R3) for any 2 lt 119904 lt 6

It follows that

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 997888rarr 0

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 997888rarr 0

(79)

As (67) we have

int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

119906

2

119899(119910) sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

= 119862

119867119871119878

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

3

119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)997888rarr 0 119899 997888rarr infin

(80)

Combining (79) and (80) with (57) yields 119906119899 rarr 0 This

contradicts inf119899|119906

119899| gt 0 Therefore case (a) cannot occur

As case (b) therefore occurs 119908119899= 119906

119899(sdot + 119886

119899) satisfies 119908

119899

119906

0= 0 From (1 + 119908

119899) minus 119869

1015840

120582(119908

119899)

1198831015840 = (1 + 119906

119899)

minus 119869

1015840

120582(119906

119899)

1198831015840 rarr 0 and the weakly sequential continuity of

minus119869

1015840

120582 we have that minus1198691015840

120582(119906

0) = 0 Therefore 119906

0is a nontrivial

solution of (1) This completes the proof


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous referees fortheir comments and suggestions on the paper Shaowei Chenwas supported by Science Foundation of Huaqiao UniversityandPromotion Program forYoung andMiddle-AgedTeacherin Science and Technology Research of Huaqiao University(ZQN-PY119)

References

[1] M Reed and B Simon Methods of Modern MathematicalPhysics IV Analysis of Operators Academic Press New YorkNY USA 1978

[2] E H Lieb ldquoThomas-Fermi and related theories of atoms andmoleculesrdquo Reviews of Modern Physics vol 53 no 4 pp 603ndash641 1981

[3] C Le Bris ldquoSome results on the Thomas-Fermi-Dirac-vonWeizsacker modelrdquo Differential and Integral Equations vol 6no 2 pp 337ndash353 1993

[4] J Lu and F Otto ldquoNonexistence of a minimizer for Thomas-Fermi-Dirac-von Weizsacker modelrdquo Communications on Pureand Applied Mathematics vol 67 no 10 pp 1605ndash1617 2014

[5] I Catto C Le Bris and P-L Lions ldquoOn some periodic Hartree-typemodels for crystalsrdquoAnnales de lrsquoInstitutHenri Poincare (C)Non Linear Analysis vol 19 no 2 pp 143ndash190 2002

[6] C Le Bris and P-L Lions ldquoFrom atoms to crystals a mathe-matical journeyrdquo Bulletin of the AmericanMathematical SocietyNew Series vol 42 no 3 pp 291ndash363 2005

[7] R Benguria H Brezis and E H Lieb ldquoTheThomas-Fermi-vonWeizsacker theory of atoms andmoleculesrdquoCommunications inMathematical Physics vol 79 no 2 pp 167ndash180 1981

[8] C O Alves M A Souto and S H Soares ldquoSchrodinger-Poisson equations without Ambrosetti-Rabinowitz conditionrdquoJournal of Mathematical Analysis and Applications vol 377 no2 pp 584ndash592 2011

[9] S Chen and C Wang ldquoExistence of multiple nontrivial solu-tions for a Schrodinger-Poisson systemrdquo Journal of Mathemati-cal Analysis and Applications vol 411 no 2 pp 787ndash793 2014

[10] T DrsquoAprile and D Mugnai ldquoSolitary waves for nonlinear Klein-Gordon-MAXwell and Schrodinger-MAXwell equationsrdquo Pro-ceedings of the Royal Society of Edinburgh Section A vol 134 no5 pp 893ndash906 2004

[11] Z Liu S Guo andZ Zhang ldquoExistence andmultiplicity of solu-tions for a class of sublinear Schrodinger-Maxwell equationsrdquoTaiwanese Journal of Mathematics vol 17 no 3 pp 857ndash8722013

[12] D Ruiz ldquoThe Schrodinger-Poisson equation under the effect ofa nonlinear local termrdquo Journal of Functional Analysis vol 237no 2 pp 655ndash674 2006

[13] Z Wang and H-S Zhou ldquoPositive solution for a nonlinearstationary Schrodinger-Poisson system in 1198773rdquo Discrete andContinuous Dynamical Systems Series A vol 18 no 4 pp 809ndash816 2007

[14] M H Yang and Z Q Han ldquoExistence and multiplicity resultsfor the nonlinear Schrodinger-Poisson systemsrdquo Nonlinear

Analysis Real World Applications vol 13 no 3 pp 1093ndash11012012

[15] L Zhao and F Zhao ldquoOn the existence of solutions forthe Schrodinger-Poisson equationsrdquo Journal of MathematicalAnalysis and Applications vol 346 no 1 pp 155ndash169 2008

[16] A Ambrosetti and D Ruiz ldquoMultiple bound states for theSchrodinger-Poisson problemrdquo Communications in Contempo-rary Mathematics vol 10 no 3 pp 391ndash404 2008

[17] A Azzollini and A Pomponio ldquoGround state solutions for thenonlinear Schrodinger-Maxwell equationsrdquo Journal of Mathe-matical Analysis and Applications vol 345 no 1 pp 90ndash1082008

[18] S Chen and L Xiao ldquoExistence of multiple nontrivial solutionsfor a strongly indefinite Schrodinger-Poisson systemrdquo Abstractand Applied Analysis vol 2014 Article ID 240208 8 pages 2014

[19] S J Chen and C L Tang ldquoHigh energy solutions for thesuperlinear Schrodinger-Maxwell equationsrdquo Nonlinear Anal-ysis Theory Methods amp Applications vol 71 no 10 pp 4927ndash4934 2009

[20] S Cingolani S Secchi and M Squassina ldquoSemi-classical limitfor Schrodinger equations withmagnetic field andHartree-typenonlinearitiesrdquo Proceedings of the Royal Society of EdinburghSection A Mathematics vol 140 no 5 pp 973ndash1009 2010

[21] M Clapp and D Salazar ldquoPositive and sign changing solutionsto a nonlinear Choquard equationrdquo Journal of MathematicalAnalysis and Applications vol 407 no 1 pp 1ndash15 2013

[22] G M Coclite ldquoA multiplicity result for the nonlinearSchrodinger-Maxwell equationsrdquo Communications in AppliedAnalysis vol 7 no 2-3 pp 417ndash423 2003

[23] M Willem Minimax Theorems Progress in Nonlinear Differ-ential Equations and Their Applications Birkhauser BostonMass USA 1996

[24] S Chen andCWang ldquoAn infinite-dimensional linking theoremwithout upper semi-continuous assumption and its applica-tionsrdquo Journal of Mathematical Analysis and Applications vol420 no 2 pp 1552ndash1567 2014

[25] L C Evans Partial Differential Equations vol 19 of GraduateStudies in Mathematics American Mathematical Society Prov-idence RI USA 1998

[26] E H Lieb ldquoSharp constants in the Hardy-Littlewood-Sobolevand related inequalitiesrdquo Annals of Mathematics vol 118 no 2pp 349ndash374 1983

[27] D Ruiz ldquoOn the Schrodinger-Poisson-Slater System behaviorof minimizers radial and nonradial casesrdquo Archive for RationalMechanics and Analysis vol 198 no 1 pp 349ndash368 2010

[28] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order Classics in Mathematics SpringerBerlin Germany 2001

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with strength 119862119890 In [3] Le Bris considered the minimizing

problem

inf120595isin1198671(R3)int

R31205952119889119909=120582

119864 (120595) (5)

with

119864 (120595) = int

R3

1003816

1003816

1003816

1003816

nabla120595

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881 (119909) 120595

2119889119909

+ 119888

1int

R3120595

103119889119909 minus 119888

2int

R3120595

83119889119909

+ int

R3int

R3

120595

2(119909) 120595

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909

(6)

where 1198881and 119888

2are positive constant and 119881 is the Coulomb

potential created by the atomic nuclei Le Bris proved thatthere exists a 120582

119888gt 0 such that for 0 lt 120582 lt 120582

119888 the problem

admits a minimizer Moreover if 1198882is small enough then the

minimizer is unique Using the Lagrange multiplier it is easyto see that the minimizer of (5) is a solution to

minus Δ120595 + 119881 (119909) 120595 + (120595

2⋆ |119909|

minus1) 120595 + 119888

1120595

73minus 119888

2120595

53

= 120583120595 inR3(7)

for some 120583 However in a recent paper [4] by Lu and Ottothe authors proved that for sufficiently large 120582 the variationalproblem (5) with 119881 = 0 does not have a minimizer Equation(1) with 119881 a periodic function is used to describe a Hartreemodel for crystals (see [5 6]) Moreover when 120582 = 0 (1) isoften referred to theThomas-Fermi-vonWeizsackermodel inthe literature One can see [7] and references therein Finallywe should mention that the so-called Schrodinger-Poisson-Slater equation

minusΔ119906 + 119881 (119909) 119906 + (119906

2⋆

1

|119909|

) 119906 = 120582 |119906|

119901minus2119906 in R

3 (8)

is also related to (1) and has attractedmuch attention in recentyears (see [5 7ndash22])

The variational functional for (1) is given by

119869

120582(119906) =

1

2

int

R3(|nabla119906|

2+ 119881 (119909) 119906

2) 119889119909

+

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus

120582

119901

int

R3|119906|

119901119889119909 +

1

119902

int

R3|119906|

119902119889119909 119906 isin 119867

1(R3)

(9)

In other words the critical points of 119869120582are solutions to (1)

Hence it is natural to use critical point theory to obtainsolutions to (1) Under assumption (v) the quadratic formint

R3(|nabla119906|

2+ 119881119906

2)119889119909 has infinite-dimensional positive and

negative spaces It can be shown that for sufficiently small120582 gt 0minus119869

120582has a global infinite-dimensional linking geometry

(for its definition see (64) of Willem [23]) However thedifficulty rises when the classical infinite-dimensional linkingtheorem (see [23]) is used to obtain a critical point to 119869

120582

This theorem requires the functional to satisfy some uppersemicontinuous assumption (see (63) of [23]) Howeverbecause the nonlinearity

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 +

1

119902

int

R3|119906|

119902119889119909 minus

120582

119901

int

R3|119906|

119901119889119909

(10)

is neither positive definite nor negative definite in 1198671(R3)whenever 120582 gt 0 minus119869

120582does not satisfy the upper semicontin-

uous assumption To overcome this difficulty we use a newinfinite-dimensional linking theorem in [24] to obtain a (119862)

119888

sequence (see Definition 6) for minus119869120582 We can prove that a (119862)

119888

sequence for minus119869120582is bounded in1198671(R3) if 120582 gt 0 is sufficiently

small This result is new and original Finally through theconcentration-compactness principle and the (119862)

119888sequence

a nontrivial solution to (1) is obtained Our method can beused to study more general equation like

minusΔ119906 + 119881 (119909) 119906 + (119906

2⋆

1

|119909|

) 119906 = 119891 (119909 119906) 119906 isin 119867

1(R3)

(11)

Notation 119861119903(119886) denotes an open ball of radius 119903 and center

119886 For a Banach space 119883 we denote the dual space of 119883 by119883

1015840 and denote strong and weak convergence in119883 by rarr and respectively For 120593 isin 1198621(119883R) we denote the Frechetderivative of 120593 at 119906 by 1205931015840(119906) The Gateaux derivative of 120593 isdenoted by ⟨1205931015840(119906) V⟩ for all 119906 V isin 119883 119871119902(R3) and 119871119902loc(R

3)

denote the standard 119871119902 space and the locally 119902-integrablefunction space respectively (1 le 119902 le infin) Let Ω be a domaininR119873 (119873 ge 1) 119862infin

0(Ω) is the space of infinitely differentiable

functions with compact support in Ω We use 119874(ℎ) and 119900(ℎ)to mean |119874(ℎ)| le 119862|ℎ| and 119900(ℎ)|ℎ| rarr 0 respectively

2 Variational Setting for (1)

LetD12(R3) be the Hilbert space

D12(R3) = 119906 isin 119871

6(R3) | int

R3|nabla119906|

2119889119909 lt infin (12)

with inner product

(119906 V)D12(R3) = intR3nabla119906nablaV 119889119909 (13)

For 119906 isin 1198671(R3) by the Lax-Milgram theorem the equation

minusΔ120601 = 4120587119906

2 in R3 (14)

has a unique solution 120601119906isin D12(R3) (see Proposition 22 of

[10]) And byTheorem 221 of [25] 120601119906can be expressed by

120601

119906(119909) = (119906

2⋆

1

1003816

1003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

) (119909) = int

R3

119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119910 119909 isin R3

(15)

The function 120601119906has the following properties



1003817

1003817

1003817

1003817

120601

119906

1003817

1003817

1003817

1003817D12(R3)le 119862 119906

2

1198671(R3) (16)

int

R3

1003816

1003816

1003816

1003816

nabla120601

119906

1003816

1003816

1003816

1003816

2

119889119909 = 4120587int

R3120601

119906119906

2119889119909

= 4120587int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862 119906

4

119871125(R3)le 119862 119906

4

1198671(R3)

(17)



4120587int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 le 119862 119906

4

119871125(R3)

(18)




119867


of 119869120582is

⟨119869

1015840

120582(119906) V⟩ = int

R3(nabla119906nablaV + 119881 (119909) 119906V) 119889119909

+ int

R3int

R3

119906 (119909) V (119909) 1199062 (119910)1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus 120582int

R3|119906|

119901minus2119906V 119889119909 + int

R3|119906|

119902minus2119906V 119889119909

forall119906 V isin 1198671 (R3)

(19)





120582


R3(|nabla119906|

2+119881(119909)119906

2)119889119909 One can




119883 = 119867

1(R3) (20)


12119906 sdot

|119871|



1 119899

2 119899




R3(|nabla119906|

2+ 119881119906

2) 119889119909 = (119906 119906) = 119906

2


R3(|nabla119906|

2+ 119881119906

2) 119889119909 = minus (119906 119906) = minus 119906

2

(21)


int

R3(|nabla119906|

2+ 119881119906

2) 119889119909 = 119875119906

2minus 119876119906

2 forall119906 isin 119883 (22)


119906 = 119875119906 + 119876119906 119906

2= 119875119906

2+ 119876119906

2 forall119906 isin 119883

(23)

By (22) and (9)

minus119869

120582(119906) =

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+

120582

119901

int

R3|119906|

119901119889119909 minus

1

119902

int

R3|119906|

119902119889119909 119906 isin 119883

(24)

Moreover by (19)

⟨minus119869

1015840

120582(119906) V⟩ = (119876119906 V) minus (119875119906 V)

minus int

R3int

R3

119906 (119909) V (119909) 1199062 (119910)1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ 120582int

R3|119906|

119901minus2119906V 119889119909

minus int

R3|119906|


(25)



|119906| = max119876119906 infin

sum

119896=1

1

2

119896+1

1003816

1003816

1003816

1003816

(119875119906 119890

119896)

1003816

1003816

1003816

1003816

(26)


0 = 1 set

119873 = 119906 isin 119885 | 119906 = 119903

119872 = 119906 + 119905119906

0| 119906 isin 119884 119905 ge 0

1003817

1003817

1003817

1003817

119906 + 119905119906

0

1003817

1003817

1003817

1003817

le 119877

120597119872 = 119906 isin 119884 | 119906 le 119877

cup 119906 + 119905119906

0| 119906 isin 119884 119905 gt 0

1003817

1003817

1003817

1003817

119906 + 119905119906

0

1003817

1003817

1003817

1003817

= 119877

(27)



119899 sub 119883 are such that 119906

119899 119906


1015840(119906) 120593⟩



120582 gt 0

sup119872

(minus119869

120582) lt +infin (28)


0 = 1


0

inf119873

(minus119869

120582) gt maxsup

120597119872

(minus119869

120582) sup|119906|le120575

(minus119869

120582(119906)) (29)


119899 119906 as


(119876119906

119899 V) 997888rarr (119876119906 V) (119875119906

119899 V) 997888rarr (119875119906 V)


Since 119906119899 119906 implies 119906

119899rarr 119906 in 119871119904loc(R



int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899V 119889119909 997888rarr int

R3|119906|

119901minus2119906V 119889119909

int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899V 119889119909 997888rarr int

R3|119906|

119902minus2119906V 119889119909

(31)





119906in 119871119904loc(R



int

R3120601

119906119899119906

119899V 119889119909 997888rarr int

R3120601

119906119906V 119889119909 (32)


⟨minus119869

1015840

120582(119906

119899) V⟩ 997888rarr ⟨minus1198691015840


0(R3) (33)




120582) lt +infin


minus119869

120582(119906) ge

1

2

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus

1

119902

int

R3|119906|

119902119889119909

ge

1

2

119906

2minus

119862

16120587

119906

4minus

119862

1

119902

119906

119902

(34)


119871119902(R3) le

119862

1119902



inf119873

(minus119869

120582) gt

1

4

119903

2gt 0 (35)


119901

|119905|

119901minus

1

119902

|119905|

119902lt 0 if 0 lt 120582 le 1 |119905| gt Λ (36)


1199061198712(R3) le 119862

1015840119906 forall119906 isin 119883

(37)


that for any 0 lt 120582 le 12058210158400

sup|119905|leΛ

119905

minus2(

120582

119901

|119905|

119901minus

1

119902

|119905|

119902) lt

1

4119862

10158402 (38)


119906 isin 119883minus 119869

120582(119906)

=

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ int

R3(

120582

119901

|119906|

119901minus

1

119902

|119906|

119902)119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ int

119909||119906(119909)|leΛ

(

120582

119901

|119906|

119901minus

1

119902

|119906|

119902)119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+

1

4119862

10158402int

R3119906

2119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2

minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 +

1

4

119906

2

=

3

4

119876119906

2minus

1

4

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

(39)



int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

ge 119862

lowast(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

(40)



0 = 1 And let 119906 = V + 119905119906

0isin


119871

2(R3 |119909|


0 (41)


any 119906 = V + 1199051199060isin 119884 oplusR119906

0

(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

ge 119862

lowastlowast119905

4

(42)


0+ V isin 119884 oplusR119906

0

minus119869

120582(119906) le

3

4

119905

2minus

1

4

V2 minus 1205811199054 (43)

It follows that

minus119869



minus119869

120582(119906) le minus

1

4

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 le 0

(45)


sup120597119872

(minus119869

120582) le 0 lt inf

119873

(minus119869

120582) (46)


0

minus119869

120582(119906) le

3

4

119876119906

2le

3

4

|119906|

2

(47)


sup|119906|le120575

(minus119869

120582(119906)) lt inf

119873

(minus119869

120582) (48)





a (119862)119888sequence for 120601 if

sup119899

120601 (119906

119899) le 119888 (1 +

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

)

1003817

1003817

1003817

1003817

1003817

120601

1015840(119906

119899)

1003817

1003817

1003817

1003817

10038171198831015840997888rarr 0

as 119899 997888rarr infin(49)





|119905|

119902minus2minus 120582 |119905|

119901minus2gt

1003817

1003817

1003817

1003817

119881

minus

1003817

1003817

1003817

1003817119871infin(R3)

if 0 lt 120582 le 1 |119905| gt 1198632

(50)



119888sequence


lim119899rarrinfin

int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (51)



V119899isin 119883 and V

119899 le 119906


that 119906119899 is a (119862)


120582 this implies ⟨1198691015840(119906

119899) V119899⟩ =


119899in (19)



119909 | 119906

119899(119909) gt 1198632 we have

119900 (1) = ⟨119869

1015840(119906

119899) V

119899⟩

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881 (119909) 119906

119899V119899119889119909

+ int

R3120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

R3(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

) 119906

119899V119899119889119909

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881

+119906

119899V119899119889119909

+ int

119909|119906119899(119909)gt1198632

120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

119909|119906119899(119909)gt1198632

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

minus 119881

minus(119909) ) 119906

119899V119899119889119909

ge int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 ge

119862(int

R3

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

=

119862(int

119909|119906119899(119909)gt1198632

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

(52)

where 119881+= 119881 + 119881



1198992 on 119909 | 119906

119899(119909) gt 119863 Together with (52)

this implies

lim119899rarrinfin

int

119909|119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (53)


lim119899rarrinfin

int

119909|minus119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (54)



0and

119906

119899 is a (119862)


120582 then 119906


Proof By (1 + 119906119899) minus 119869

1015840

120582(119906

119899)

1198831015840 rarr 0 we have

⟨minus119869

1015840

120582(119906

119899) 119876119906

119899⟩ = 119900 (1) ⟨minus119869

1015840

120582(119906

119899) 119875119906

119899⟩ = 119900 (1)

(55)


Choosing V = 119876119906119899and V = 119875119906


1003817

1003817

1003817

1003817

119876119906

119896

1003817

1003817

1003817

1003817

2

= int

R3119906

119899120601

119906119899sdot 119876119906

119899119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 + 119900 (1)

1003817

1003817

1003817

1003817

119875119906

119899

1003817

1003817

1003817

1003817

2

= minus int

R3119906

119899120601

119906119899sdot 119875119906

119899119889119909

+ int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 + 119900 (1)

(56)


2= 119875119906

119899

2+ 119876119906

119899


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

(57)

Let

120594

1198991(119909) =

1 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

le 119863

0 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

gt 119863

(58)

and 1205941198992= 1 minus 120594


119899=

(119906

119899120594

1198991)

2+ (119906

119899120594

1198992)

2 and

120601

119906119899= 120601

1199061198991205941198991+ 120601

1199061198991205941198992 (59)

Since 1206011199061198991205941198991


minusΔ120601 = 4120587 (119906

119899120594

1198991)

2 in R3

(60)

by (119906119899120594

1198991)

2le 119863




1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(1198611(119910))

le 119862

2(int

1198612(119910)

1003816

1003816

1003816

1003816

119906

119899120594

1198991

1003816

1003816

1003816

1003816

4

119889119909)

12

le 119862

2119863

2(int

1198612(119910)

119889119909)

12

(61)

Let 1198623= (int

1198612(0)119889119909)

12 By (61)

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(R3)le 119862

2119862

3119863

2 (62)

Together with 1206011199061198991205941198991


0 le 120601

1199061198991205941198991le 119862

2119862

3119863

2 in R3 (63)

By (57) (59) and (63) we have

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

1199061198991205941198991sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|gt119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

le

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991119906

119899

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ (int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ 120582(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119901(R3)

+ (int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119902(R3)

(64)

Since 119876119906119899minus 119875119906

119899

1198712(R3) le 2119862

1015840119906

119899 (1198621015840 is the constant


119899

119871119901(R3) le 119862

10158401015840119906

119899 and

119876119906

119899minus 119875119906

119899

119871119902(R3) le 119862

10158401015840119906



1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840 10038171003817

1003817

1003817

1003817

119906

119899120601

1199061198991205941198991

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 120582119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(65)



|119906

119899|

119901119889119909 = 119900(1) and

int

119909||119906119899(119909)|gt119863|119906

119899|



1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(66)


int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

(119906

119899120594

1198992(119910))

2

sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

le 119862

101584010158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909)

56

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

(67)

where 119862119867119871119878


int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909 = 119900 (1) if 0 lt 120582 le 1 (68)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(69)


(int

119909||119906119899(119909)|le120598

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

le (int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus1

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus1

)

2

119889119909)

12

le 2(int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119901minus2

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119902minus2

) 119889119909)

12

le 4 (120598

119901minus2+ 120598

119902minus2) (int

119909||119906119899(119909)|le120598

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909)

12

le 4119862

1015840(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(70)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 4119862

10158402(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909|120598lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(71)


0+

120598

119902minus2


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 4119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 4119862

1015840(int

119909|1205980lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(72)


120582(119906

119899)) le 119888 and (1+119906

119899)minus119869

1015840

120582(119906

119899)

1198831015840 rarr 0

we obtain

119900 (1) + 119888 ge minus119869

120582(119906

119899) +

1

2

⟨119869

1015840

120582(119906

119899) 119906

119899⟩

=

1

4

int

R3119906

2

119899120601

119906119899119889119909

minus int

R3(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus (

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

(73)


It follows that

1

4

int

R3119906

2

119899120601

119906119899119889119909

+ int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + int

119909||119906119899(119909)|le1205821(119902minus119901)

(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus(

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

+ 119900 (1)

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

(74)



0

int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

(75)


2

119899120601

119906119899119889119909 ge int

R3119906

2

119899120601

1199061198991205941198991119889119909 =

119906

119899120601

12

1199061198991205941198991

2

1198712(R3)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(76)



02

and 1205980le |119905| le 119863 then

119862

5(120582 |119905|

119901minus2119905 minus |119905|

119902minus2119905)

2

le (

1

2

minus

1

119902

) |119905|

119902minus 120582(

1

2

minus

1

119901

) |119905|

119901

(77)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ 119862

5int

119909|119863ge|119906119899(119909)|gt1205980

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(78)


(0 120598

119902minus119901



0 lt 120582 lt 120582

10158401015840

0



0 120582

10158401015840

0 where 1205821015840

0and 12058210158401015840

0



119888sequence 119906

119899 for minus119869

120582

with inf119899|119906



sup119910isinR3 int1198611(119910)

|119906

119899|

2119889119909 = 0 or


int

1198611(119886119899)|119906

119899|

2119889119909 ge 984858




It follows that

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 997888rarr 0

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 997888rarr 0

(79)

As (67) we have

int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

119906

2

119899(119910) sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

= 119862

119867119871119878

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

3

119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)997888rarr 0 119899 997888rarr infin

(80)





119899(sdot + 119886

119899) satisfies 119908

119899

119906

0= 0 From (1 + 119908

119899) minus 119869

1015840

120582(119908

119899)

1198831015840 = (1 + 119906

119899)

minus 119869

1015840

120582(119906

119899)


minus119869

1015840


120582(119906

0) = 0 Therefore 119906

0is a nontrivial





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817

1003817

1003817

1003817

120601

119906

1003817

1003817

1003817

1003817D12(R3)le 119862 119906

2

1198671(R3) (16)

int

R3

1003816

1003816

1003816

1003816

nabla120601

119906

1003816

1003816

1003816

1003816

2

119889119909 = 4120587int

R3120601

119906119906

2119889119909

= 4120587int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862 119906

4

119871125(R3)le 119862 119906

4

1198671(R3)

(17)



4120587int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 le 119862 119906

4

119871125(R3)

(18)




119867


of 119869120582is

⟨119869

1015840

120582(119906) V⟩ = int

R3(nabla119906nablaV + 119881 (119909) 119906V) 119889119909

+ int

R3int

R3

119906 (119909) V (119909) 1199062 (119910)1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus 120582int

R3|119906|

119901minus2119906V 119889119909 + int

R3|119906|

119902minus2119906V 119889119909

forall119906 V isin 1198671 (R3)

(19)





120582


R3(|nabla119906|

2+119881(119909)119906

2)119889119909 One can




119883 = 119867

1(R3) (20)


12119906 sdot

|119871|



1 119899

2 119899




R3(|nabla119906|

2+ 119881119906

2) 119889119909 = (119906 119906) = 119906

2


R3(|nabla119906|

2+ 119881119906

2) 119889119909 = minus (119906 119906) = minus 119906

2

(21)


int

R3(|nabla119906|

2+ 119881119906

2) 119889119909 = 119875119906

2minus 119876119906

2 forall119906 isin 119883 (22)


119906 = 119875119906 + 119876119906 119906

2= 119875119906

2+ 119876119906

2 forall119906 isin 119883

(23)

By (22) and (9)

minus119869

120582(119906) =

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+

120582

119901

int

R3|119906|

119901119889119909 minus

1

119902

int

R3|119906|

119902119889119909 119906 isin 119883

(24)

Moreover by (19)

⟨minus119869

1015840

120582(119906) V⟩ = (119876119906 V) minus (119875119906 V)

minus int

R3int

R3

119906 (119909) V (119909) 1199062 (119910)1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ 120582int

R3|119906|

119901minus2119906V 119889119909

minus int

R3|119906|


(25)



|119906| = max119876119906 infin

sum

119896=1

1

2

119896+1

1003816

1003816

1003816

1003816

(119875119906 119890

119896)

1003816

1003816

1003816

1003816

(26)


0 = 1 set

119873 = 119906 isin 119885 | 119906 = 119903

119872 = 119906 + 119905119906

0| 119906 isin 119884 119905 ge 0

1003817

1003817

1003817

1003817

119906 + 119905119906

0

1003817

1003817

1003817

1003817

le 119877

120597119872 = 119906 isin 119884 | 119906 le 119877

cup 119906 + 119905119906

0| 119906 isin 119884 119905 gt 0

1003817

1003817

1003817

1003817

119906 + 119905119906

0

1003817

1003817

1003817

1003817

= 119877

(27)



119899 sub 119883 are such that 119906

119899 119906


1015840(119906) 120593⟩



120582 gt 0

sup119872

(minus119869

120582) lt +infin (28)


0 = 1


0

inf119873

(minus119869

120582) gt maxsup

120597119872

(minus119869

120582) sup|119906|le120575

(minus119869

120582(119906)) (29)


119899 119906 as


(119876119906

119899 V) 997888rarr (119876119906 V) (119875119906

119899 V) 997888rarr (119875119906 V)


Since 119906119899 119906 implies 119906

119899rarr 119906 in 119871119904loc(R



int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899V 119889119909 997888rarr int

R3|119906|

119901minus2119906V 119889119909

int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899V 119889119909 997888rarr int

R3|119906|

119902minus2119906V 119889119909

(31)





119906in 119871119904loc(R



int

R3120601

119906119899119906

119899V 119889119909 997888rarr int

R3120601

119906119906V 119889119909 (32)


⟨minus119869

1015840

120582(119906

119899) V⟩ 997888rarr ⟨minus1198691015840


0(R3) (33)




120582) lt +infin


minus119869

120582(119906) ge

1

2

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus

1

119902

int

R3|119906|

119902119889119909

ge

1

2

119906

2minus

119862

16120587

119906

4minus

119862

1

119902

119906

119902

(34)


119871119902(R3) le

119862

1119902



inf119873

(minus119869

120582) gt

1

4

119903

2gt 0 (35)


119901

|119905|

119901minus

1

119902

|119905|

119902lt 0 if 0 lt 120582 le 1 |119905| gt Λ (36)


1199061198712(R3) le 119862

1015840119906 forall119906 isin 119883

(37)


that for any 0 lt 120582 le 12058210158400

sup|119905|leΛ

119905

minus2(

120582

119901

|119905|

119901minus

1

119902

|119905|

119902) lt

1

4119862

10158402 (38)


119906 isin 119883minus 119869

120582(119906)

=

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ int

R3(

120582

119901

|119906|

119901minus

1

119902

|119906|

119902)119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ int

119909||119906(119909)|leΛ

(

120582

119901

|119906|

119901minus

1

119902

|119906|

119902)119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+

1

4119862

10158402int

R3119906

2119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2

minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 +

1

4

119906

2

=

3

4

119876119906

2minus

1

4

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

(39)



int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

ge 119862

lowast(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

(40)



0 = 1 And let 119906 = V + 119905119906

0isin


119871

2(R3 |119909|


0 (41)


any 119906 = V + 1199051199060isin 119884 oplusR119906

0

(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

ge 119862

lowastlowast119905

4

(42)


0+ V isin 119884 oplusR119906

0

minus119869

120582(119906) le

3

4

119905

2minus

1

4

V2 minus 1205811199054 (43)

It follows that

minus119869



minus119869

120582(119906) le minus

1

4

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 le 0

(45)


sup120597119872

(minus119869

120582) le 0 lt inf

119873

(minus119869

120582) (46)


0

minus119869

120582(119906) le

3

4

119876119906

2le

3

4

|119906|

2

(47)


sup|119906|le120575

(minus119869

120582(119906)) lt inf

119873

(minus119869

120582) (48)





a (119862)119888sequence for 120601 if

sup119899

120601 (119906

119899) le 119888 (1 +

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

)

1003817

1003817

1003817

1003817

1003817

120601

1015840(119906

119899)

1003817

1003817

1003817

1003817

10038171198831015840997888rarr 0

as 119899 997888rarr infin(49)





|119905|

119902minus2minus 120582 |119905|

119901minus2gt

1003817

1003817

1003817

1003817

119881

minus

1003817

1003817

1003817

1003817119871infin(R3)

if 0 lt 120582 le 1 |119905| gt 1198632

(50)



119888sequence


lim119899rarrinfin

int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (51)



V119899isin 119883 and V

119899 le 119906


that 119906119899 is a (119862)


120582 this implies ⟨1198691015840(119906

119899) V119899⟩ =


119899in (19)



119909 | 119906

119899(119909) gt 1198632 we have

119900 (1) = ⟨119869

1015840(119906

119899) V

119899⟩

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881 (119909) 119906

119899V119899119889119909

+ int

R3120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

R3(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

) 119906

119899V119899119889119909

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881

+119906

119899V119899119889119909

+ int

119909|119906119899(119909)gt1198632

120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

119909|119906119899(119909)gt1198632

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

minus 119881

minus(119909) ) 119906

119899V119899119889119909

ge int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 ge

119862(int

R3

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

=

119862(int

119909|119906119899(119909)gt1198632

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

(52)

where 119881+= 119881 + 119881



1198992 on 119909 | 119906

119899(119909) gt 119863 Together with (52)

this implies

lim119899rarrinfin

int

119909|119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (53)


lim119899rarrinfin

int

119909|minus119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (54)



0and

119906

119899 is a (119862)


120582 then 119906


Proof By (1 + 119906119899) minus 119869

1015840

120582(119906

119899)

1198831015840 rarr 0 we have

⟨minus119869

1015840

120582(119906

119899) 119876119906

119899⟩ = 119900 (1) ⟨minus119869

1015840

120582(119906

119899) 119875119906

119899⟩ = 119900 (1)

(55)


Choosing V = 119876119906119899and V = 119875119906


1003817

1003817

1003817

1003817

119876119906

119896

1003817

1003817

1003817

1003817

2

= int

R3119906

119899120601

119906119899sdot 119876119906

119899119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 + 119900 (1)

1003817

1003817

1003817

1003817

119875119906

119899

1003817

1003817

1003817

1003817

2

= minus int

R3119906

119899120601

119906119899sdot 119875119906

119899119889119909

+ int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 + 119900 (1)

(56)


2= 119875119906

119899

2+ 119876119906

119899


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

(57)

Let

120594

1198991(119909) =

1 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

le 119863

0 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

gt 119863

(58)

and 1205941198992= 1 minus 120594


119899=

(119906

119899120594

1198991)

2+ (119906

119899120594

1198992)

2 and

120601

119906119899= 120601

1199061198991205941198991+ 120601

1199061198991205941198992 (59)

Since 1206011199061198991205941198991


minusΔ120601 = 4120587 (119906

119899120594

1198991)

2 in R3

(60)

by (119906119899120594

1198991)

2le 119863




1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(1198611(119910))

le 119862

2(int

1198612(119910)

1003816

1003816

1003816

1003816

119906

119899120594

1198991

1003816

1003816

1003816

1003816

4

119889119909)

12

le 119862

2119863

2(int

1198612(119910)

119889119909)

12

(61)

Let 1198623= (int

1198612(0)119889119909)

12 By (61)

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(R3)le 119862

2119862

3119863

2 (62)

Together with 1206011199061198991205941198991


0 le 120601

1199061198991205941198991le 119862

2119862

3119863

2 in R3 (63)

By (57) (59) and (63) we have

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

1199061198991205941198991sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|gt119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

le

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991119906

119899

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ (int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ 120582(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119901(R3)

+ (int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119902(R3)

(64)

Since 119876119906119899minus 119875119906

119899

1198712(R3) le 2119862

1015840119906

119899 (1198621015840 is the constant


119899

119871119901(R3) le 119862

10158401015840119906

119899 and

119876119906

119899minus 119875119906

119899

119871119902(R3) le 119862

10158401015840119906



1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840 10038171003817

1003817

1003817

1003817

119906

119899120601

1199061198991205941198991

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 120582119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(65)



|119906

119899|

119901119889119909 = 119900(1) and

int

119909||119906119899(119909)|gt119863|119906

119899|



1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(66)


int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

(119906

119899120594

1198992(119910))

2

sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

le 119862

101584010158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909)

56

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

(67)

where 119862119867119871119878


int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909 = 119900 (1) if 0 lt 120582 le 1 (68)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(69)


(int

119909||119906119899(119909)|le120598

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

le (int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus1

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus1

)

2

119889119909)

12

le 2(int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119901minus2

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119902minus2

) 119889119909)

12

le 4 (120598

119901minus2+ 120598

119902minus2) (int

119909||119906119899(119909)|le120598

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909)

12

le 4119862

1015840(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(70)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 4119862

10158402(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909|120598lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(71)


0+

120598

119902minus2


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 4119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 4119862

1015840(int

119909|1205980lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(72)


120582(119906

119899)) le 119888 and (1+119906

119899)minus119869

1015840

120582(119906

119899)

1198831015840 rarr 0

we obtain

119900 (1) + 119888 ge minus119869

120582(119906

119899) +

1

2

⟨119869

1015840

120582(119906

119899) 119906

119899⟩

=

1

4

int

R3119906

2

119899120601

119906119899119889119909

minus int

R3(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus (

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

(73)


It follows that

1

4

int

R3119906

2

119899120601

119906119899119889119909

+ int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + int

119909||119906119899(119909)|le1205821(119902minus119901)

(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus(

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

+ 119900 (1)

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

(74)



0

int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

(75)


2

119899120601

119906119899119889119909 ge int

R3119906

2

119899120601

1199061198991205941198991119889119909 =

119906

119899120601

12

1199061198991205941198991

2

1198712(R3)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(76)



02

and 1205980le |119905| le 119863 then

119862

5(120582 |119905|

119901minus2119905 minus |119905|

119902minus2119905)

2

le (

1

2

minus

1

119902

) |119905|

119902minus 120582(

1

2

minus

1

119901

) |119905|

119901

(77)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ 119862

5int

119909|119863ge|119906119899(119909)|gt1205980

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(78)


(0 120598

119902minus119901



0 lt 120582 lt 120582

10158401015840

0



0 120582

10158401015840

0 where 1205821015840

0and 12058210158401015840

0



119888sequence 119906

119899 for minus119869

120582

with inf119899|119906



sup119910isinR3 int1198611(119910)

|119906

119899|

2119889119909 = 0 or


int

1198611(119886119899)|119906

119899|

2119889119909 ge 984858




It follows that

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 997888rarr 0

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 997888rarr 0

(79)

As (67) we have

int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

119906

2

119899(119910) sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

= 119862

119867119871119878

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

3

119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)997888rarr 0 119899 997888rarr infin

(80)





119899(sdot + 119886

119899) satisfies 119908

119899

119906

0= 0 From (1 + 119908

119899) minus 119869

1015840

120582(119908

119899)

1198831015840 = (1 + 119906

119899)

minus 119869

1015840

120582(119906

119899)


minus119869

1015840


120582(119906

0) = 0 Therefore 119906

0is a nontrivial





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899 sub 119883 are such that 119906

119899 119906


1015840(119906) 120593⟩



120582 gt 0

sup119872

(minus119869

120582) lt +infin (28)


0 = 1


0

inf119873

(minus119869

120582) gt maxsup

120597119872

(minus119869

120582) sup|119906|le120575

(minus119869

120582(119906)) (29)


119899 119906 as


(119876119906

119899 V) 997888rarr (119876119906 V) (119875119906

119899 V) 997888rarr (119875119906 V)


Since 119906119899 119906 implies 119906

119899rarr 119906 in 119871119904loc(R



int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899V 119889119909 997888rarr int

R3|119906|

119901minus2119906V 119889119909

int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899V 119889119909 997888rarr int

R3|119906|

119902minus2119906V 119889119909

(31)





119906in 119871119904loc(R



int

R3120601

119906119899119906

119899V 119889119909 997888rarr int

R3120601

119906119906V 119889119909 (32)


⟨minus119869

1015840

120582(119906

119899) V⟩ 997888rarr ⟨minus1198691015840


0(R3) (33)




120582) lt +infin


minus119869

120582(119906) ge

1

2

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

minus

1

119902

int

R3|119906|

119902119889119909

ge

1

2

119906

2minus

119862

16120587

119906

4minus

119862

1

119902

119906

119902

(34)


119871119902(R3) le

119862

1119902



inf119873

(minus119869

120582) gt

1

4

119903

2gt 0 (35)


119901

|119905|

119901minus

1

119902

|119905|

119902lt 0 if 0 lt 120582 le 1 |119905| gt Λ (36)


1199061198712(R3) le 119862

1015840119906 forall119906 isin 119883

(37)


that for any 0 lt 120582 le 12058210158400

sup|119905|leΛ

119905

minus2(

120582

119901

|119905|

119901minus

1

119902

|119905|

119902) lt

1

4119862

10158402 (38)


119906 isin 119883minus 119869

120582(119906)

=

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ int

R3(

120582

119901

|119906|

119901minus

1

119902

|119906|

119902)119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+ int

119909||119906(119909)|leΛ

(

120582

119901

|119906|

119901minus

1

119902

|119906|

119902)119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

+

1

4119862

10158402int

R3119906

2119889119909

le

1

2

119876119906

2minus

1

2

119875119906

2

minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 +

1

4

119906

2

=

3

4

119876119906

2minus

1

4

119875119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

(39)



int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

ge 119862

lowast(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

(40)



0 = 1 And let 119906 = V + 119905119906

0isin


119871

2(R3 |119909|


0 (41)


any 119906 = V + 1199051199060isin 119884 oplusR119906

0

(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

ge 119862

lowastlowast119905

4

(42)


0+ V isin 119884 oplusR119906

0

minus119869

120582(119906) le

3

4

119905

2minus

1

4

V2 minus 1205811199054 (43)

It follows that

minus119869



minus119869

120582(119906) le minus

1

4

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 le 0

(45)


sup120597119872

(minus119869

120582) le 0 lt inf

119873

(minus119869

120582) (46)


0

minus119869

120582(119906) le

3

4

119876119906

2le

3

4

|119906|

2

(47)


sup|119906|le120575

(minus119869

120582(119906)) lt inf

119873

(minus119869

120582) (48)





a (119862)119888sequence for 120601 if

sup119899

120601 (119906

119899) le 119888 (1 +

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

)

1003817

1003817

1003817

1003817

1003817

120601

1015840(119906

119899)

1003817

1003817

1003817

1003817

10038171198831015840997888rarr 0

as 119899 997888rarr infin(49)





|119905|

119902minus2minus 120582 |119905|

119901minus2gt

1003817

1003817

1003817

1003817

119881

minus

1003817

1003817

1003817

1003817119871infin(R3)

if 0 lt 120582 le 1 |119905| gt 1198632

(50)



119888sequence


lim119899rarrinfin

int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (51)



V119899isin 119883 and V

119899 le 119906


that 119906119899 is a (119862)


120582 this implies ⟨1198691015840(119906

119899) V119899⟩ =


119899in (19)



119909 | 119906

119899(119909) gt 1198632 we have

119900 (1) = ⟨119869

1015840(119906

119899) V

119899⟩

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881 (119909) 119906

119899V119899119889119909

+ int

R3120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

R3(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

) 119906

119899V119899119889119909

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881

+119906

119899V119899119889119909

+ int

119909|119906119899(119909)gt1198632

120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

119909|119906119899(119909)gt1198632

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

minus 119881

minus(119909) ) 119906

119899V119899119889119909

ge int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 ge

119862(int

R3

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

=

119862(int

119909|119906119899(119909)gt1198632

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

(52)

where 119881+= 119881 + 119881



1198992 on 119909 | 119906

119899(119909) gt 119863 Together with (52)

this implies

lim119899rarrinfin

int

119909|119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (53)


lim119899rarrinfin

int

119909|minus119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (54)



0and

119906

119899 is a (119862)


120582 then 119906


Proof By (1 + 119906119899) minus 119869

1015840

120582(119906

119899)

1198831015840 rarr 0 we have

⟨minus119869

1015840

120582(119906

119899) 119876119906

119899⟩ = 119900 (1) ⟨minus119869

1015840

120582(119906

119899) 119875119906

119899⟩ = 119900 (1)

(55)


Choosing V = 119876119906119899and V = 119875119906


1003817

1003817

1003817

1003817

119876119906

119896

1003817

1003817

1003817

1003817

2

= int

R3119906

119899120601

119906119899sdot 119876119906

119899119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 + 119900 (1)

1003817

1003817

1003817

1003817

119875119906

119899

1003817

1003817

1003817

1003817

2

= minus int

R3119906

119899120601

119906119899sdot 119875119906

119899119889119909

+ int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 + 119900 (1)

(56)


2= 119875119906

119899

2+ 119876119906

119899


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

(57)

Let

120594

1198991(119909) =

1 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

le 119863

0 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

gt 119863

(58)

and 1205941198992= 1 minus 120594


119899=

(119906

119899120594

1198991)

2+ (119906

119899120594

1198992)

2 and

120601

119906119899= 120601

1199061198991205941198991+ 120601

1199061198991205941198992 (59)

Since 1206011199061198991205941198991


minusΔ120601 = 4120587 (119906

119899120594

1198991)

2 in R3

(60)

by (119906119899120594

1198991)

2le 119863




1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(1198611(119910))

le 119862

2(int

1198612(119910)

1003816

1003816

1003816

1003816

119906

119899120594

1198991

1003816

1003816

1003816

1003816

4

119889119909)

12

le 119862

2119863

2(int

1198612(119910)

119889119909)

12

(61)

Let 1198623= (int

1198612(0)119889119909)

12 By (61)

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(R3)le 119862

2119862

3119863

2 (62)

Together with 1206011199061198991205941198991


0 le 120601

1199061198991205941198991le 119862

2119862

3119863

2 in R3 (63)

By (57) (59) and (63) we have

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

1199061198991205941198991sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|gt119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

le

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991119906

119899

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ (int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ 120582(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119901(R3)

+ (int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119902(R3)

(64)

Since 119876119906119899minus 119875119906

119899

1198712(R3) le 2119862

1015840119906

119899 (1198621015840 is the constant


119899

119871119901(R3) le 119862

10158401015840119906

119899 and

119876119906

119899minus 119875119906

119899

119871119902(R3) le 119862

10158401015840119906



1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840 10038171003817

1003817

1003817

1003817

119906

119899120601

1199061198991205941198991

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 120582119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(65)



|119906

119899|

119901119889119909 = 119900(1) and

int

119909||119906119899(119909)|gt119863|119906

119899|



1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(66)


int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

(119906

119899120594

1198992(119910))

2

sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

le 119862

101584010158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909)

56

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

(67)

where 119862119867119871119878


int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909 = 119900 (1) if 0 lt 120582 le 1 (68)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(69)


(int

119909||119906119899(119909)|le120598

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

le (int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus1

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus1

)

2

119889119909)

12

le 2(int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119901minus2

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119902minus2

) 119889119909)

12

le 4 (120598

119901minus2+ 120598

119902minus2) (int

119909||119906119899(119909)|le120598

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909)

12

le 4119862

1015840(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(70)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 4119862

10158402(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909|120598lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(71)


0+

120598

119902minus2


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 4119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 4119862

1015840(int

119909|1205980lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(72)


120582(119906

119899)) le 119888 and (1+119906

119899)minus119869

1015840

120582(119906

119899)

1198831015840 rarr 0

we obtain

119900 (1) + 119888 ge minus119869

120582(119906

119899) +

1

2

⟨119869

1015840

120582(119906

119899) 119906

119899⟩

=

1

4

int

R3119906

2

119899120601

119906119899119889119909

minus int

R3(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus (

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

(73)


It follows that

1

4

int

R3119906

2

119899120601

119906119899119889119909

+ int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + int

119909||119906119899(119909)|le1205821(119902minus119901)

(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus(

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

+ 119900 (1)

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

(74)



0

int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

(75)


2

119899120601

119906119899119889119909 ge int

R3119906

2

119899120601

1199061198991205941198991119889119909 =

119906

119899120601

12

1199061198991205941198991

2

1198712(R3)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(76)



02

and 1205980le |119905| le 119863 then

119862

5(120582 |119905|

119901minus2119905 minus |119905|

119902minus2119905)

2

le (

1

2

minus

1

119902

) |119905|

119902minus 120582(

1

2

minus

1

119901

) |119905|

119901

(77)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ 119862

5int

119909|119863ge|119906119899(119909)|gt1205980

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(78)


(0 120598

119902minus119901



0 lt 120582 lt 120582

10158401015840

0



0 120582

10158401015840

0 where 1205821015840

0and 12058210158401015840

0



119888sequence 119906

119899 for minus119869

120582

with inf119899|119906



sup119910isinR3 int1198611(119910)

|119906

119899|

2119889119909 = 0 or


int

1198611(119886119899)|119906

119899|

2119889119909 ge 984858




It follows that

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 997888rarr 0

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 997888rarr 0

(79)

As (67) we have

int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

119906

2

119899(119910) sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

= 119862

119867119871119878

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

3

119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)997888rarr 0 119899 997888rarr infin

(80)





119899(sdot + 119886

119899) satisfies 119908

119899

119906

0= 0 From (1 + 119908

119899) minus 119869

1015840

120582(119908

119899)

1198831015840 = (1 + 119906

119899)

minus 119869

1015840

120582(119906

119899)


minus119869

1015840


120582(119906

0) = 0 Therefore 119906

0is a nontrivial





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 = 1 And let 119906 = V + 119905119906

0isin


119871

2(R3 |119909|


0 (41)


any 119906 = V + 1199051199060isin 119884 oplusR119906

0

(int

R3

119906

2

|119909|

12(1 + |ln |119909||)

119889119909)

2

ge 119862

lowastlowast119905

4

(42)


0+ V isin 119884 oplusR119906

0

minus119869

120582(119906) le

3

4

119905

2minus

1

4

V2 minus 1205811199054 (43)

It follows that

minus119869



minus119869

120582(119906) le minus

1

4

119906

2minus

1

4

int

R3int

R3

119906

2(119909) 119906

2(119910)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910 le 0

(45)


sup120597119872

(minus119869

120582) le 0 lt inf

119873

(minus119869

120582) (46)


0

minus119869

120582(119906) le

3

4

119876119906

2le

3

4

|119906|

2

(47)


sup|119906|le120575

(minus119869

120582(119906)) lt inf

119873

(minus119869

120582) (48)





a (119862)119888sequence for 120601 if

sup119899

120601 (119906

119899) le 119888 (1 +

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

)

1003817

1003817

1003817

1003817

1003817

120601

1015840(119906

119899)

1003817

1003817

1003817

1003817

10038171198831015840997888rarr 0

as 119899 997888rarr infin(49)





|119905|

119902minus2minus 120582 |119905|

119901minus2gt

1003817

1003817

1003817

1003817

119881

minus

1003817

1003817

1003817

1003817119871infin(R3)

if 0 lt 120582 le 1 |119905| gt 1198632

(50)



119888sequence


lim119899rarrinfin

int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (51)



V119899isin 119883 and V

119899 le 119906


that 119906119899 is a (119862)


120582 this implies ⟨1198691015840(119906

119899) V119899⟩ =


119899in (19)



119909 | 119906

119899(119909) gt 1198632 we have

119900 (1) = ⟨119869

1015840(119906

119899) V

119899⟩

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881 (119909) 119906

119899V119899119889119909

+ int

R3120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

R3(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

) 119906

119899V119899119889119909

= int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 + int

R3119881

+119906

119899V119899119889119909

+ int

119909|119906119899(119909)gt1198632

120601

119906119899sdot 119906

119899sdot V119899119889119909

+ int

119909|119906119899(119909)gt1198632

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

minus 120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

minus 119881

minus(119909) ) 119906

119899V119899119889119909

ge int

R3

1003816

1003816

1003816

1003816

nablaV119899

1003816

1003816

1003816

1003816

2

119889119909 ge

119862(int

R3

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

=

119862(int

119909|119906119899(119909)gt1198632

1003816

1003816

1003816

1003816

V119899

1003816

1003816

1003816

1003816

6

119889119909)

13

(52)

where 119881+= 119881 + 119881



1198992 on 119909 | 119906

119899(119909) gt 119863 Together with (52)

this implies

lim119899rarrinfin

int

119909|119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (53)


lim119899rarrinfin

int

119909|minus119906119899(119909)gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

6

119889119909 = 0 (54)



0and

119906

119899 is a (119862)


120582 then 119906


Proof By (1 + 119906119899) minus 119869

1015840

120582(119906

119899)

1198831015840 rarr 0 we have

⟨minus119869

1015840

120582(119906

119899) 119876119906

119899⟩ = 119900 (1) ⟨minus119869

1015840

120582(119906

119899) 119875119906

119899⟩ = 119900 (1)

(55)


Choosing V = 119876119906119899and V = 119875119906


1003817

1003817

1003817

1003817

119876119906

119896

1003817

1003817

1003817

1003817

2

= int

R3119906

119899120601

119906119899sdot 119876119906

119899119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 + 119900 (1)

1003817

1003817

1003817

1003817

119875119906

119899

1003817

1003817

1003817

1003817

2

= minus int

R3119906

119899120601

119906119899sdot 119875119906

119899119889119909

+ int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 + 119900 (1)

(56)


2= 119875119906

119899

2+ 119876119906

119899


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

(57)

Let

120594

1198991(119909) =

1 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

le 119863

0 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

gt 119863

(58)

and 1205941198992= 1 minus 120594


119899=

(119906

119899120594

1198991)

2+ (119906

119899120594

1198992)

2 and

120601

119906119899= 120601

1199061198991205941198991+ 120601

1199061198991205941198992 (59)

Since 1206011199061198991205941198991


minusΔ120601 = 4120587 (119906

119899120594

1198991)

2 in R3

(60)

by (119906119899120594

1198991)

2le 119863




1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(1198611(119910))

le 119862

2(int

1198612(119910)

1003816

1003816

1003816

1003816

119906

119899120594

1198991

1003816

1003816

1003816

1003816

4

119889119909)

12

le 119862

2119863

2(int

1198612(119910)

119889119909)

12

(61)

Let 1198623= (int

1198612(0)119889119909)

12 By (61)

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(R3)le 119862

2119862

3119863

2 (62)

Together with 1206011199061198991205941198991


0 le 120601

1199061198991205941198991le 119862

2119862

3119863

2 in R3 (63)

By (57) (59) and (63) we have

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

1199061198991205941198991sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|gt119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

le

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991119906

119899

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ (int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ 120582(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119901(R3)

+ (int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119902(R3)

(64)

Since 119876119906119899minus 119875119906

119899

1198712(R3) le 2119862

1015840119906

119899 (1198621015840 is the constant


119899

119871119901(R3) le 119862

10158401015840119906

119899 and

119876119906

119899minus 119875119906

119899

119871119902(R3) le 119862

10158401015840119906



1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840 10038171003817

1003817

1003817

1003817

119906

119899120601

1199061198991205941198991

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 120582119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(65)



|119906

119899|

119901119889119909 = 119900(1) and

int

119909||119906119899(119909)|gt119863|119906

119899|



1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(66)


int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

(119906

119899120594

1198992(119910))

2

sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

le 119862

101584010158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909)

56

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

(67)

where 119862119867119871119878


int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909 = 119900 (1) if 0 lt 120582 le 1 (68)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(69)


(int

119909||119906119899(119909)|le120598

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

le (int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus1

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus1

)

2

119889119909)

12

le 2(int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119901minus2

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119902minus2

) 119889119909)

12

le 4 (120598

119901minus2+ 120598

119902minus2) (int

119909||119906119899(119909)|le120598

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909)

12

le 4119862

1015840(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(70)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 4119862

10158402(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909|120598lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(71)


0+

120598

119902minus2


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 4119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 4119862

1015840(int

119909|1205980lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(72)


120582(119906

119899)) le 119888 and (1+119906

119899)minus119869

1015840

120582(119906

119899)

1198831015840 rarr 0

we obtain

119900 (1) + 119888 ge minus119869

120582(119906

119899) +

1

2

⟨119869

1015840

120582(119906

119899) 119906

119899⟩

=

1

4

int

R3119906

2

119899120601

119906119899119889119909

minus int

R3(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus (

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

(73)


It follows that

1

4

int

R3119906

2

119899120601

119906119899119889119909

+ int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + int

119909||119906119899(119909)|le1205821(119902minus119901)

(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus(

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

+ 119900 (1)

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

(74)



0

int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

(75)


2

119899120601

119906119899119889119909 ge int

R3119906

2

119899120601

1199061198991205941198991119889119909 =

119906

119899120601

12

1199061198991205941198991

2

1198712(R3)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(76)



02

and 1205980le |119905| le 119863 then

119862

5(120582 |119905|

119901minus2119905 minus |119905|

119902minus2119905)

2

le (

1

2

minus

1

119902

) |119905|

119902minus 120582(

1

2

minus

1

119901

) |119905|

119901

(77)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ 119862

5int

119909|119863ge|119906119899(119909)|gt1205980

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(78)


(0 120598

119902minus119901



0 lt 120582 lt 120582

10158401015840

0



0 120582

10158401015840

0 where 1205821015840

0and 12058210158401015840

0



119888sequence 119906

119899 for minus119869

120582

with inf119899|119906



sup119910isinR3 int1198611(119910)

|119906

119899|

2119889119909 = 0 or


int

1198611(119886119899)|119906

119899|

2119889119909 ge 984858




It follows that

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 997888rarr 0

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 997888rarr 0

(79)

As (67) we have

int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

119906

2

119899(119910) sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

= 119862

119867119871119878

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

3

119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)997888rarr 0 119899 997888rarr infin

(80)





119899(sdot + 119886

119899) satisfies 119908

119899

119906

0= 0 From (1 + 119908

119899) minus 119869

1015840

120582(119908

119899)

1198831015840 = (1 + 119906

119899)

minus 119869

1015840

120582(119906

119899)


minus119869

1015840


120582(119906

0) = 0 Therefore 119906

0is a nontrivial





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Choosing V = 119876119906119899and V = 119875119906


1003817

1003817

1003817

1003817

119876119906

119896

1003817

1003817

1003817

1003817

2

= int

R3119906

119899120601

119906119899sdot 119876119906

119899119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 + 119900 (1)

1003817

1003817

1003817

1003817

119875119906

119899

1003817

1003817

1003817

1003817

2

= minus int

R3119906

119899120601

119906119899sdot 119875119906

119899119889119909

+ int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 + 119900 (1)

(56)


2= 119875119906

119899

2+ 119876119906

119899


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

(57)

Let

120594

1198991(119909) =

1 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

le 119863

0 if 100381610038161003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

gt 119863

(58)

and 1205941198992= 1 minus 120594


119899=

(119906

119899120594

1198991)

2+ (119906

119899120594

1198992)

2 and

120601

119906119899= 120601

1199061198991205941198991+ 120601

1199061198991205941198992 (59)

Since 1206011199061198991205941198991


minusΔ120601 = 4120587 (119906

119899120594

1198991)

2 in R3

(60)

by (119906119899120594

1198991)

2le 119863




1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(1198611(119910))

le 119862

2(int

1198612(119910)

1003816

1003816

1003816

1003816

119906

119899120594

1198991

1003816

1003816

1003816

1003816

4

119889119909)

12

le 119862

2119863

2(int

1198612(119910)

119889119909)

12

(61)

Let 1198623= (int

1198612(0)119889119909)

12 By (61)

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817119871infin(R3)le 119862

2119862

3119863

2 (62)

Together with 1206011199061198991205941198991


0 le 120601

1199061198991205941198991le 119862

2119862

3119863

2 in R3 (63)

By (57) (59) and (63) we have

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

= int

R3119906

119899sdot 120601

1199061198991205941198991sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909

minus int

119909||119906119899(119909)|gt119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

sdot (119876119906

119899minus 119875119906

119899) 119889119909 + 119900 (1)

le

1003817

1003817

1003817

1003817

1003817

120601

1199061198991205941198991119906

119899

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ (int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

10038171198712(R3)

+ 120582(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119901(R3)

+ (int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871119902(R3)

(64)

Since 119876119906119899minus 119875119906

119899

1198712(R3) le 2119862

1015840119906

119899 (1198621015840 is the constant


119899

119871119901(R3) le 119862

10158401015840119906

119899 and

119876119906

119899minus 119875119906

119899

119871119902(R3) le 119862

10158401015840119906



1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840 10038171003817

1003817

1003817

1003817

119906

119899120601

1199061198991205941198991

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 120582119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

119889119909)

(119901minus1)119901

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119862

10158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

119889119909)

(119902minus1)119902

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(65)



|119906

119899|

119901119889119909 = 119900(1) and

int

119909||119906119899(119909)|gt119863|119906

119899|



1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(66)


int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

(119906

119899120594

1198992(119910))

2

sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

le 119862

101584010158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909)

56

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

(67)

where 119862119867119871119878


int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909 = 119900 (1) if 0 lt 120582 le 1 (68)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(69)


(int

119909||119906119899(119909)|le120598

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

le (int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus1

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus1

)

2

119889119909)

12

le 2(int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119901minus2

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119902minus2

) 119889119909)

12

le 4 (120598

119901minus2+ 120598

119902minus2) (int

119909||119906119899(119909)|le120598

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909)

12

le 4119862

1015840(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(70)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 4119862

10158402(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909|120598lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(71)


0+

120598

119902minus2


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 4119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 4119862

1015840(int

119909|1205980lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(72)


120582(119906

119899)) le 119888 and (1+119906

119899)minus119869

1015840

120582(119906

119899)

1198831015840 rarr 0

we obtain

119900 (1) + 119888 ge minus119869

120582(119906

119899) +

1

2

⟨119869

1015840

120582(119906

119899) 119906

119899⟩

=

1

4

int

R3119906

2

119899120601

119906119899119889119909

minus int

R3(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus (

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

(73)


It follows that

1

4

int

R3119906

2

119899120601

119906119899119889119909

+ int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + int

119909||119906119899(119909)|le1205821(119902minus119901)

(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus(

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

+ 119900 (1)

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

(74)



0

int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

(75)


2

119899120601

119906119899119889119909 ge int

R3119906

2

119899120601

1199061198991205941198991119889119909 =

119906

119899120601

12

1199061198991205941198991

2

1198712(R3)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(76)



02

and 1205980le |119905| le 119863 then

119862

5(120582 |119905|

119901minus2119905 minus |119905|

119902minus2119905)

2

le (

1

2

minus

1

119902

) |119905|

119902minus 120582(

1

2

minus

1

119901

) |119905|

119901

(77)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ 119862

5int

119909|119863ge|119906119899(119909)|gt1205980

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(78)


(0 120598

119902minus119901



0 lt 120582 lt 120582

10158401015840

0



0 120582

10158401015840

0 where 1205821015840

0and 12058210158401015840

0



119888sequence 119906

119899 for minus119869

120582

with inf119899|119906



sup119910isinR3 int1198611(119910)

|119906

119899|

2119889119909 = 0 or


int

1198611(119886119899)|119906

119899|

2119889119909 ge 984858




It follows that

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 997888rarr 0

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 997888rarr 0

(79)

As (67) we have

int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

119906

2

119899(119910) sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

= 119862

119867119871119878

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

3

119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)997888rarr 0 119899 997888rarr infin

(80)





119899(sdot + 119886

119899) satisfies 119908

119899

119906

0= 0 From (1 + 119908

119899) minus 119869

1015840

120582(119908

119899)

1198831015840 = (1 + 119906

119899)

minus 119869

1015840

120582(119906

119899)


minus119869

1015840


120582(119906

0) = 0 Therefore 119906

0is a nontrivial





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











|119906

119899|

119901119889119909 = 119900(1) and

int

119909||119906119899(119909)|gt119863|119906

119899|



1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(66)


int

R3119906

119899sdot 120601

1199061198991205941198992sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

(119906

119899120594

1198992(119910))

2

sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

(119906

119899120594

1198992)

210038171003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

sdot

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

le 119862

101584010158401015840(int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909)

56

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

(67)

where 119862119867119871119878


int

119909||119906119899(119909)|gt119863

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

125

119889119909 = 119900 (1) if 0 lt 120582 le 1 (68)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909||119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(69)


(int

119909||119906119899(119909)|le120598

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

le (int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus1

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus1

)

2

119889119909)

12

le 2(int

119909||119906119899(119909)|le120598

(

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119901minus2

+

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2119902minus2

) 119889119909)

12

le 4 (120598

119901minus2+ 120598

119902minus2) (int

119909||119906119899(119909)|le120598

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909)

12

le 4119862

1015840(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(70)


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 2119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 4119862

10158402(120598

119901minus2+ 120598

119902minus2)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 2119862

1015840(int

119909|120598lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(71)


0+

120598

119902minus2


1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

le 4119862

1015840119863(119862

2119862

3)

121003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

10038171198712(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1) sdot

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

+ 4119862

1015840(int

119909|1205980lt|119906119899(119909)|le119863

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899

minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909)

12

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

(72)


120582(119906

119899)) le 119888 and (1+119906

119899)minus119869

1015840

120582(119906

119899)

1198831015840 rarr 0

we obtain

119900 (1) + 119888 ge minus119869

120582(119906

119899) +

1

2

⟨119869

1015840

120582(119906

119899) 119906

119899⟩

=

1

4

int

R3119906

2

119899120601

119906119899119889119909

minus int

R3(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus (

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

(73)


It follows that

1

4

int

R3119906

2

119899120601

119906119899119889119909

+ int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + int

119909||119906119899(119909)|le1205821(119902minus119901)

(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus(

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

+ 119900 (1)

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

(74)



0

int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

(75)


2

119899120601

119906119899119889119909 ge int

R3119906

2

119899120601

1199061198991205941198991119889119909 =

119906

119899120601

12

1199061198991205941198991

2

1198712(R3)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(76)



02

and 1205980le |119905| le 119863 then

119862

5(120582 |119905|

119901minus2119905 minus |119905|

119902minus2119905)

2

le (

1

2

minus

1

119902

) |119905|

119902minus 120582(

1

2

minus

1

119901

) |119905|

119901

(77)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ 119862

5int

119909|119863ge|119906119899(119909)|gt1205980

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(78)


(0 120598

119902minus119901



0 lt 120582 lt 120582

10158401015840

0



0 120582

10158401015840

0 where 1205821015840

0and 12058210158401015840

0



119888sequence 119906

119899 for minus119869

120582

with inf119899|119906



sup119910isinR3 int1198611(119910)

|119906

119899|

2119889119909 = 0 or


int

1198611(119886119899)|119906

119899|

2119889119909 ge 984858




It follows that

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 997888rarr 0

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 997888rarr 0

(79)

As (67) we have

int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

119906

2

119899(119910) sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

= 119862

119867119871119878

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

3

119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)997888rarr 0 119899 997888rarr infin

(80)





119899(sdot + 119886

119899) satisfies 119908

119899

119906

0= 0 From (1 + 119908

119899) minus 119869

1015840

120582(119908

119899)

1198831015840 = (1 + 119906

119899)

minus 119869

1015840

120582(119906

119899)


minus119869

1015840


120582(119906

0) = 0 Therefore 119906

0is a nontrivial





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










It follows that

1

4

int

R3119906

2

119899120601

119906119899119889119909

+ int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899(119909)

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + int

119909||119906119899(119909)|le1205821(119902minus119901)

(120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

minus(

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

)119889119909

+ 119900 (1)

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

(74)



0

int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le int

119909||119906119899(119909)|gt1205821(119902minus119901)

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

(75)


2

119899120601

119906119899119889119909 ge int

R3119906

2

119899120601

1199061198991205941198991119889119909 =

119906

119899120601

12

1199061198991205941198991

2

1198712(R3)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ int

119909|119863ge|119906119899(119909)|gt1205980

((

1

2

minus

1

119902

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902

minus 120582(

1

2

minus

1

119901

)

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901

)119889119909

le 119888 + 119862

4120582

(119902minus2)(119902minus119901)int

R3

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

2

119889119909 + 119900 (1)

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(76)



02

and 1205980le |119905| le 119863 then

119862

5(120582 |119905|

119901minus2119905 minus |119905|

119902minus2119905)

2

le (

1

2

minus

1

119902

) |119905|

119902minus 120582(

1

2

minus

1

119901

) |119905|

119901

(77)


1

4

1003817

1003817

1003817

1003817

1003817

1003817

119906

119899120601

12

1199061198991205941198991

1003817

1003817

1003817

1003817

1003817

1003817

2

1198712(R3)

+ 119862

5int

119909|119863ge|119906119899(119909)|gt1205980

(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899)

2

119889119909

le 119888 + 119862

10158402119862

4120582

(119902minus2)(119902minus119901) 10038171003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

2

+ 119900 (1)

(78)


(0 120598

119902minus119901



0 lt 120582 lt 120582

10158401015840

0



0 120582

10158401015840

0 where 1205821015840

0and 12058210158401015840

0



119888sequence 119906

119899 for minus119869

120582

with inf119899|119906



sup119910isinR3 int1198611(119910)

|119906

119899|

2119889119909 = 0 or


int

1198611(119886119899)|119906

119899|

2119889119909 ge 984858




It follows that

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119876119906

119899119889119909 997888rarr 0

int

R3(120582

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119901minus2

119906

119899minus

1003816

1003816

1003816

1003816

119906

119899

1003816

1003816

1003816

1003816

119902minus2

119906

119899) sdot 119875119906

119899119889119909 997888rarr 0

(79)

As (67) we have

int

R3119906

119899sdot 120601

119906119899sdot (119876119906

119899minus 119875119906

119899) 119889119909

= int

R3int

R3

119906

2

119899(119910) sdot 119906

119899(119909) sdot (119876119906

119899minus 119875119906

119899) (119909)

1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119889119909 119889119910

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899sdot (119876119906

119899minus 119875119906

119899)

1003817

1003817

1003817

100381711987165(R3)

le 119862

119867119871119878

1003817

1003817

1003817

1003817

1003817

119906

2

119899

1003817

1003817

1003817

1003817

100381711987165(R3)

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)

= 119862

119867119871119878

1003817

1003817

1003817

1003817

119906

119899

1003817

1003817

1003817

1003817

3

119871125(R3)

1003817

1003817

1003817

1003817

119876119906

119899minus 119875119906

119899

1003817

1003817

1003817

1003817119871125(R3)997888rarr 0 119899 997888rarr infin

(80)





119899(sdot + 119886

119899) satisfies 119908

119899

119906

0= 0 From (1 + 119908

119899) minus 119869

1015840

120582(119908

119899)

1198831015840 = (1 + 119906

119899)

minus 119869

1015840

120582(119906

119899)


minus119869

1015840


120582(119906

0) = 0 Therefore 119906

0is a nontrivial





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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