Multiple positive solutions for a Hamiltonian system with an indefinite data function

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Multiple positive solutions for aHamiltonian system with an indefinitedata functionBrahim Khaldi a & Nasreddine Megrez ba Faculty of Sciences and Technology , University of Bechar , PB417, Bechar 8000 , Algeriab Chemical and Material Engineering Department , University ofAlberta , 9107 - 116 Street, Edmonton , AB T6G 2V4 , CanadaPublished online: 20 Sep 2011.

To cite this article: Brahim Khaldi & Nasreddine Megrez (2013) Multiple positive solutions for aHamiltonian system with an indefinite data function, Complex Variables and Elliptic Equations: AnInternational Journal, 58:5, 699-713, DOI: 10.1080/17476933.2011.609929

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Multiple positive solutions for a Hamiltonian system with an

indefinite data function

Brahim Khaldia* and Nasreddine Megrezb

aFaculty of Sciences and Technology, University of Bechar, PB 417, Bechar 8000, Algeria;bChemical and Material Engineering Department, University of Alberta, 9107 - 116 Street,

Edmonton, AB T6G 2V4 Canada

Communicated by T. Bartsch

(Received 13 December 2010; final version received 25 July 2011)

In this article, using the sub-supersolution method and Li–Willem’stheorem, we establish the existence of multiple solutions for a nonhomo-geneous subcritical Hamiltonian system with a data function allowed tochange sign. A nonexistence result is provided as well.

Keywords: Hamiltonian system; sub-supersolution method; multiplicity ofsolutions; linking geometry; strongly indefinite functionals; Li–Willemtheorem

AMS Subject Classifications: 35J65; 35J20; 35J47

1. Introduction

In this article, we study the existence of multiple solutions for the following system:

ðS�Þ

�Du ¼ vp in �

�Dv ¼ uq þ �f ðxÞ in �

u4 0, v4 0 in �

u ¼ v ¼ 0 on @�,

8>>><>>>:where � is a bounded regular domain of R

N, (N� 3) with smooth boundary @�,p, q41 such that 1

pþ1þ1

qþ1 4 1� 2N, � is a positive parameter and f� 0 satisfies:

( f1) f is a continuous function allowed to change sign,( f2) The linear Dirichlet problem

�Dv ¼ f ðxÞ in �

v ¼ 0 on @�

�ð1Þ

has nonnegative solution denoted by vf.

*Corresponding author. Email: [email protected]

� 201 Taylor & Francis

http://dx.doi.org/10.1080/17476933.2011.609929

Complex Variables and Elliptic Equations, 2013

Vol. 58, No. 5, 699–713,

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For the existence of a function f (x) that changes sign and satisfies ( f2),

we refer to [1].We call the point ( p, q) critical if

1

pþ 1þ

1

qþ 1¼ 1�

2

N,

subcritical if

1

pþ 1þ

1

qþ 14 1�

2

N

and supercritical if

1

pþ 1þ

1

qþ 15 1�

2

N:

Much attention has been paid to the existence of nontrivial solutions for the

following system:

ðS Þ

�Du ¼@H

@vðx, u, vÞ in �

�Dv ¼@H

@uðx, u, vÞ in �

u ¼ v ¼ 0 on @�,

8>>>><>>>>:where H is a C1 real function defined on ��R� R:

De Figueiredo and Felmer [2] considered the subcritical case for the system (S )

and proved that under:

(H1) H : �� R2! R is a C1 function,

(H2) There are p, q40 and a positive constant C such that

Hðx, u, vÞ � C uj jpþ1þ vj jqþ1þ1� �

for all x, u, vð Þ 2��R2,

(H3) There exists a positive constant C such that

@H

@uðx, u, vÞ

�� C uj jpþ vj jpðqþ1Þpþ1

� �@H

@vðx, u, vÞ

�� C uj jqð pþ1Þpþ1 þ vj jq

� �for all x, u, vð Þ 2��R

2,

(H4) There exists R40 such that

1

pþ 1

@H

@uðx, u, vÞ þ

1

qþ 1

@H

@vðx, u, vÞ � Hðx, u, vÞ4 0 for all x2� and ðu, vÞ

�� R,

(H5) There exist r40 and c40 such that

Hðx, u, vÞ�� c

�uj jpþ1þ vj jqþ1

�for all x2� and ðu, vÞ

�� r,

(H6) p, q4 0 such that 141

pþ 1þ

1

qþ 14 1�

2

N,

2 B. Khaldi and N. Megrez700

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the system (S ) has a strong solution in the sense that, if (u, v) 2 H10ð�Þ

� �2then

(u, v) 2 ½W1,pþ1p0 ð�Þ \W

2,pþ1p0 ð�Þ� � ½W

1,qþ1q0 ð�Þ \W

2,qþ1q0 ð�Þ�. A similar result was

obtained by Hulshoff and Van der Vorst [3] (see also [4]), proving that (S) has atleast one nontrivial solution by means of the direct min–max method of Benci andRabinowitz [5], and using fractional Sobolev spaces.

Dai and Gu [6], studied the single equation problem

�Du ¼ uq þ �f ðxÞ in �u4 0 in �u ¼ 0 on @�

8<: ð2Þ

where � is a bounded domain of RN, and f� 0 satisfies ( f 1) and ( f 2).

Han and Liu [7] showed that the problem

ðS�,�,�,f,gÞ

�Du ¼ �vþ vp þ �gðxÞ in �

�Dv ¼ �uþ uq þ �f ðxÞ in �

u4 0, v4 0 in �

u ¼ v ¼ 0 on @�:

8>>><>>>:has a minimal solution for �, �2 [0,�1), p, q41 (Theorem 1.1 in [7]), where �1 is thefirst eigenvalue of the Laplacian; and has a second solution for �, �2 ]0,�1) in thecritical and subcritical cases (Theorem 1.3 in [7]). f, g are two functions satisfying ( f1)and ( f 2) with f� 0 and g� 0.

In our work, we consider the subcritical case and we establish the existence ofmultiple solutions for (S�,�,�,f,g) even when �¼ �¼ 0 and either f or g is the zerofunction. Without loss of generality, we take g� 0 reducing the problem to the formof (S�) as a coupling version of Equation (2) with �Du¼ up. One solution at least isobtained by sub-supersolution method, and another one at least is obtained byapplying Li–Willem’s theorem [8].

This article is organized as follows. Section 2 deals with the existence of aminimal positive solution. Section 3 is devoted to the proof of the existence of asecond positive solution, and finally, Section 4 provides a non-existence result.

Our main results are stated as follows.

THEOREM 1.1 Assume that p41, q41 and f satisfies ( f1) and ( f2). Then there exists apositive number �0 such that (S�) has a minimal solution (u�, v�) for each �2 (0, �0).

THEOREM 1.2 Let N� 3, p, q41 and 1pþ1þ

1qþ1 4 1� 2

N. Then for all �2 (0, �0),the problem (S�) admits a second solution ðbu,bv Þ such that bu4 u� and bv4 v�:

THEOREM 1.3 Let �*¼ supf�02Rþj (S�) has at least one solution for �2 (0, �0)}.

Under the hypothesis of Theorem 1.2, there exists �*4�* such that problem (S�)has no solution for �4�*.

2. Existence of a minimal solution

In this section, we prove the Theorem 1.1 where the main statement is the existenceof a minimal solution (u�, v�) of (S�). A solution (u�, v�) is said to be minimal if anyother solution (u, v) of (S�) satisfies u� u� and v� v� in �.

Complex Variables and Elliptic Equations 3701

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Proof of Theorem 1.1 For 1� r�1, we denote by j�jr the Lr-norm on the Lebesgue

spaces Lr(�).Let vf be the nonnegative solution given by the assumption ( f2) and set v¼ �vf,

where � is a positive parameter. Then we have

�Dv ¼ �f ðxÞ � �f ðxÞ þ uq for all u � 0:

Consider the following problem:

ðSgÞ�Du ¼ gðxÞð Þ

p in �

u ¼ 0 on @�,

�where 0 � gðxÞ � minðvðxÞ,M

1pÞ and M is a positive constant to be specified later.

By Lax–Milgram lemma, (Sg) admits a unique solution denoted by ug. Moreover by

maximum principle, ug40 in �.Put u¼ ug, then we have

�Du � vp for all v � v4 0:

Hence (u, v) is a subsolution of system (S�).Now, we construct a supersolution of (S�) as u, vð Þ ¼ ðMe,NeÞ, where M and N

are positive constants and e is the positive solution of the problem

�De ¼ 1 in �e ¼ 0 on @�:

�The pair u, vð Þ is a supersolution of (S�) if M and N satisfy

M ¼ Np ej jp1 and N ¼Mq ej jq1þ � f�� 1:

The coordinates of the maximum point of the curve

�ðMÞ ¼1

f�� 1

ej j1M

1p �Mq ej jqþ11

� �are M0 ¼ pq ej jqþ11

� p1�pq and �10 :¼ �ðM0Þ ¼

ð pq�1Þjej

qð pþ1Þ1�pq1

j f j1 pqð Þpq

pq�14 0 since pq41. The strong

maximum principle applied to

�Dðu� uÞ ¼M� gðxÞð Þp� 0 in �

u� u ¼ 0 on @�:

�and

�Dðv� vÞ ¼ N� �f ðxÞ � 0 in �

v� v ¼ 0 on @�,

�for � � �20 :¼ M

1p

ej j1 fj j1with M5M0, ensures that u4 u and v4 v:

Hence, we have at least a solution (u, v) for the system

�Du ¼ vp in ��Dv ¼ uq þ �f ðxÞ in �u ¼ v ¼ 0 on @�,

8<:


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such that

05 ug � u �Me and 0 � �vf � v � Ne for � � �0 ¼ minð�10, �20Þ:

We see that u40 since ug40. But, vf is nonnegative. So, to show that v40 we apply

the maximum principle to

�Dðv� vÞ ¼ uq � 0 in �

v� v ¼ 0 on @�,

�to get v4v� 0.

Hence, (S�) has at least a solution (u, v).Now, we prove the existence of a minimal solution (u�, v�) by using the

monotone iteration scheme and sub and super-solution method [9]. Let (u, v) be any

solution of (S�). In fact, (u, v) can be considered as a supersolution of (S�).

Furthermore, u4u and v4v in �, where (u, v) is the subsolution described

previously. Indeed, the system

�Dðu� uÞ ¼ vp � gðxÞð Þp� vp � vp 4 0 in �

�Dðv� vÞ ¼ uq 4 0 in �

u� u ¼ v� v ¼ 0: on @�

8><>:combined with the maximum principle yield that u4u and v4v in �.

Now, by the monotone iteration scheme

u0 :¼ u, v0 :¼ v

n ¼ 0, 1, 2, . . .

�Dunþ1 ¼ vpn in �

�Dvnþ1 ¼ uqn þ �f xð Þ in �

unþ1 4 0, vnþ1 4 0 in �

unþ1 ¼ vnþ1 ¼ 0 on @�,

8>>>>>>>><>>>>>>>>:and the strong maximum principle, we obtain a sequence (un, vn) of subsolutions of

(S�) such that

05 u0 5 u1 5 u2 5 � � � 5 un 5 � � � 5 u in �

and

0 � v0 5 v1 5 v2 5 � � � 5 vn 5 � � � 5 v in �:

Let us denote the limit of (un, vn) by (u�, v�):

un % u� and vn % v� in �:

Then, (u�, v�) is a solution of (S�). Furthermore, 05u�� u and 05v�� v in �, which

imply that (u�, v�) is a minimal solution of (S�). g


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3. Existence of a second solution

In this section, we prove Theorem 1.2. We look for a second solution bu,bvð Þ of (S�)under the form (u�þ u, v�þ v) for all �2 (0, �0), where (u�, v�) denotes the minimalsolution of (S�) described in the previous section.

The corresponding system for (u, v) is

ðeS�Þ�Du ¼ ðvþ v�Þ

p� v

p� in �

�Dv ¼ ðuþ u�Þq� u

q� in �

u4 0, v4 0 in �u ¼ v ¼ 0 on @�:

8>><>>:with p, q41 and 14 1

pþ1þ1

qþ1 4 1� 2N :

Let E be a Hilbert space and I : E!R be a strongly indefinite functionalnear zero in the sense that there exist two subspaces Eþ and E� with E¼EþE�

such that the functional I is positive definite on Eþ and negative definite on E�

(near zero). We also assume that there are sequences of subspaces of finitedimensions En such that

E1 � E2 � E3 . . . , and [1n¼1En ¼ E:

Denote

En ¼ Eþn E�n , and In ¼ IjEn :

We have

E1 � E2 � E3 . . . , and [1n¼1En ¼ E:

Definition 3.1 We say that I satisfies the (PS*) condition with respect to the scale ofsubspaces (En)n if every sequence (zn)n such that

zn 2En, InðznÞ�� C, I0nðzkÞ, �

�� "n �k kE, for all �2En, and "n! 0,

contains a subsequence which converges to a critical point of I.

We need the following result of Li and Willem [9].

THEOREM 3.2 Let �2C1(E,R) such that:

(A1) � has a local linking at the origin, i.e. for some r40

�ðzÞ � 0 for z2Eþ, and �ðzÞ � 0 for z2E�, with zk kE� r,

(A2) � maps bounded sets into bounded sets,(A3) �(z)!�1 as kzk!1, z2Eþn E�, for every n2N,(A4) � satisfies the (PS*) condition with respect to the scale of subspaces (En)n.Then, � has a nontrivial critical point.

3.1. The variational formulation of ðeSkÞ

In order to set up our problem variationally, we shall use fractional Sobolev spacesthat we define using Fourier series. For more details we refer to the article of DeFigueiredo [10].


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Definition 3.3 Let (�n) be the eigenvalues of ð�D,H10ð�ÞÞ and (’n) be the

corresponding eigenfunctions normalized by j’nj2¼ 1.For s� 0 we define

Es ¼ u2L2ð�Þ :X1n¼1

�sna2n 51, where u ¼

X1n¼1

an’n

( )and the operator

As : Es�!L2ð�Þ such that Asu ¼X1n¼1

�s2nan’n

Es is a Hilbert space with the inner product given by

u, vh iEs¼X1n¼1

�snanbn for v ¼X1n¼1

bn’n and u ¼X1n¼1

an’n:

Observe that

u, vh iEs¼

Z�

AsuAsv dx:

So, As is an isometric isomorphism and its inverse is denoted by A�s.The Sobolev imbedding theorem for the spaces Es says that Es

�Lrcontinuously

if 1r �

12�

sN, and compactly if the previous inequality is strict.

From the inequality 14 1pþ1þ

1qþ1 4 1� 2

N , we can choose s, t40 such that

sþ t¼ 2, t4s and

1

pþ 14

1

2�

s

N,

1

qþ 14

1

2�

t

N:

Thus, Es�Lpþ1(�) and E t

�Lqþ1(�) with compact immersions.Now, let E¼E t

�Es. We see that for z¼ (u, v)2E, the functional

IðzÞ ¼

Z�

AtuAsv dx�

Z�

FðuÞ dx�

Z�

GðvÞ dx,

with

FðuÞ ¼1

qþ 1uþ u�j jqþ1�u

qþ1�

� �� u

q�u, and GðvÞ ¼

1

pþ 1

�vþ v�j jpþ1�v

pþ1�

�� v

p�v,

is well defined and is of class C1. Its derivative is given by the following expression:Z�

�AtuAs þ At’Asv� gðvÞ � f ðuÞ’

�dx ¼ 0, for any ð’, Þ 2E, ð3Þ

where

f ðuÞ ¼ F 0ðuÞ ¼ uþ u�j jq�1 uþ u�ð Þ � uq� and gðvÞ ¼ G0ðvÞ ¼ vþ v�j jp�1 vþ v�ð Þ � v

p�:


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From the expressions of F and G we get the following properties:

limu!þ1

F uð Þ

uj jqþ1¼

1

qþ 1, lim

u!0

F uð Þ

uj j¼ 0

and

limv!þ1

GðvÞ

vj jpþ1¼

1

pþ 1, lim

v!0

GðvÞ

vj j¼ 0:

As a consequence, for "40 we obtain

F uð Þ � " uj j þ C uj jqþ1 ð4Þ

and

G vð Þ � " vj j þ C vj jpþ1:

To show the positivity of F(u) and G(v) we also need the following auxiliary result.

LEMMA 3.4 Let A, B and � be real numbers satisfying A� 0,B� 0 and �41. Then

A� Bj j�� A� � �A��1B ð5Þ

and

Aþ Bj j�� A� þ �A��1B: ð6Þ

Proof If jA�Bj ¼A (resp. jAþBj ¼A), then (5) (resp. (6)) is trivial.

So, it suffices to show (5) when jA�Bj 6¼A (i.e. jA�Bj5A or jA�Bj4A) and(6) when jAþBj4A.

We first give the proof of (5) in the case jA�Bj5A. Indeed, for D2 (jA�Bj, A)we have

A� � A� Bj j� ¼ �D��1 A� A� Bj jð Þ

� �D��1B:

The fact that �41, f (t)¼ t��1 is an increasing function on (0, þ1), we obtain

A� � A� Bj j�� A��1B:

The proof is similar when jA�Bj4A and for (6) when jAþBj4A. g

LEMMA 3.5 Let (u, v)2E¼Et�Es. Then F(u)� 0 and G(v)� 0.

Proof We consider two cases.

Case 1 u� 0.Using the inequality (6), we get

1

qþ 1uþ u�j jqþ1�

1

qþ 1uqþ1� þ u

q�u,

then

FðuÞ � 0:


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Case 2 u� 0.

FðuÞ ¼1

qþ 1uþ u�j jqþ1�u

qþ1�

� �� u

q�u

¼1

qþ 1

��u� � uj j��qþ1 � u

qþ1�

� � u

q�u:

Using (5), we obtain

1

qþ 1

��u� � uj j��qþ1 � 1

qþ 1uqþ1� � u

q� uj j,

hence

FðuÞ � 0:

Similarly, we get G(v)� 0. g

So, the critical points of the functional I given by (3) are the weak solutions

(u, v)2E t�E s of the systemZ

�

AtuAs dx ¼

Z�

g vð Þ dx, 8 2Es

Z�

At’Asv dx ¼

Z�

f uð Þ’ dx, 8’2Et:

8>>><>>>:These weak solutions (u, v) are in fact strong solutions of (S�). That is, u2W

2,qþ1q \

W1,qþ1q0 and v2W 2,pþ1p \W

1,pþ1p0 . This regularity result was proved in [2].

Following De Figueiredo and Felmer [2], we can define the space

E¼EþE� with

Eþ ¼ ðu,At�suÞ j u2Et� �

, E� ¼ ðu, �At�suÞ j u2Et� �

and

1

2zk k2E¼ QðzþÞ �Qðz�Þ,

where zþ2Eþ, z�2E� and Q(z)¼R

�AtuAsv for z¼ (u, v)2E.

Proof of Theorem 1.2 In this proof, C and " are generic constants, meaning that

they are not necessarily the same everywhere.We start by checking the conditions of Li–Willem’s theorem.First, it is clear that I(z) is a C1 functional on E t

�E s.

(A1) For z2Eþ, we have

IðzÞ ¼

Z�

Atu�� 2�FðuÞ � GðAt�suÞ� �

dx:

By using (4), we obtain

IðzÞ �

Z�

Atu�� 2dx� C

Z�

uj jqþ1dxþ

Z�

At�su�� pþ1dx�

� "

Z�

uj jdxþ

Z�

At�su�� dx�

� uk k2Et�C uj jqþ1qþ1þ At�su

�� pþ1pþ1

� �� "

�uj j1þ At�su

�� 1

�:


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Now, by Sobolev’s embedding and � :¼min{p, q}, we have

IðzÞ � uk k2Et�C uk kqþ1Et þ At�su

�� pþ1Es

� �� "

�uk kEtþ At�su

�� Es

�� uk k2Et�2C uk k�þ1Et � 2" uk kEt , for uk kEt small enough.

Then, for "! 0 and kzkE small enough, we have I(z)� 0 for z2Eþ.On the other hand, by Lemma 3.5, we find for z2E�

IðzÞ � � uk k2Et :

Hence, I(z) has a local linking with respect to the origin.

(A2) Let B�Et�Es be a bounded set, i.e. there exists C40 such that uk kEt� C and

uk kEs� C, for all (u, v)2B.Then,

IðzÞ�� Atu

�� 2Asvj j2þ

Z�

uþ u�j jqþ1dxþ

Z�

vþ v�j jpþ1dxþ u�j jq1

Z�

uj jdx

þ v�j jp1

Z�

vj jdxþ C

� uk kEt vk kEsþ2qZ

�

uj jqþ1dxþ 2pZ

�

vj jpþ1dxþ u�j jq1

Z�

uj jdxþ v�j jp1

Z�

vj jdxþ C

� uk kEt vk kEs þC 2q uk kqþ1Et þ 2p vk k

pþ1Es þ u�j j

q1 uk kEt þ v�j j

p1 vk kEs þ 1

� �� C:

Thus, I(z) maps bounded sets into bounded sets.

(A3) Let zk ¼ zþk þ z�k 2Eþn E� denote a sequence with kzkkE!1.

By the definitions of Eþ and E�, zk can be written as

zk ¼�uk þ vk,A

t�sðuk � vkÞ�, with uk 2E

tn, vk 2E

t,

where Etn denote an n-dimensional subspace of Et.

Thus, the functional I(zk) takes the from

IðzkÞ ¼

Z�

Atuk�� 2dx� Z

�

Atvk�� 2dx� Z

�

Fðuk þ vkÞdx�

Z�

G�At�sðuk � vkÞ

�dx:

Using the inequality (6) and the fact that t� s40, we have the estimatesZ�

Fðuk þ vkÞ ¼1

qþ 1

Z�

uk þ vk þ u�j jqþ1dx�

Z�

uq� uk þ vkj jdx� C

�1

qþ 1

Z�

��juk þ vkj � u�

��qþ1dx� C

Z�

uk þ vkj jqþ1� 1

qþ1

dx� C

�1

qþ 1

Z�

uk þ vkj jqþ1dx�

Z�

uk þ vkj jqu�dx� C uk þ vkj jqþ1�C

� C uk þ vkj j�þ12 � C uk þ vkk kqEt�C uk þ vkk kEt�C,


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and Z�

G�At�sðuk � vkÞ

�dx

¼

Z�

1

pþ 1At�sðuk � vkÞ þ v�� pþ1�vp� At�sðuk � vkÞ

�� dx� C

�

Z�

1

pþ 1

�� At�sðuk � vkÞ�� þ v�

��pþ1 � vp�

�� 1

At�sðuk � vkÞ��

dx� C

�

Z�

1

pþ 1At�sðuk � vkÞ�� pþ1� At�sðuk � vkÞ

�� pv�� dx� C uk � vkk kEt�C

� C

Z�

At�sðuk � vkÞ�� 2� pþ1

2

dx� v�j j1 At�sðuk � vkÞ�� p

p�C uk � vkk kEt�C

� C uk � vkj jpþ12 �C uk � vkj j

pEt�C uk � vkk kEt�C

� C uk � vkj j�þ12 � C uk � vkj jpEt�C uk � vkk kEt�C:

Hence, we obtain

IðzkÞ � Atuk�� 2

2�C uk þ vkj j�þ12 þ uk � vkj j�þ12

� �þ C uk þ vkk k

qEt

þ C uk � vkk kpEtþC ukk kEtþC vkk kEtþC

� ukk k2Et�

C

2�

�uk þ vkj j2þ uk � vkj j2

��þ1þ C

�ukk kEtþ vkk kEt

�qþ C


�pþ C ukk kEtþC vkk kEtþC

� ukk k2Et�

C

2�ukj j

�þ12 þ C


�qþ C


�pþ C ukk kEtþC vkk kEtþC:

We know that the norms �k kEt and j�j2 are equivalent on Etn. Thus,

IðzkÞ � ukk k2Et�

C

2�ukk k

�þ1Et þ C


�qþ C


�pþ C ukk kEtþC vkk kEtþC:

ð7Þ

Note that

zkk kE�!1() ukk k2Etþ vkk k

2Et �!1:

(a) If ukk kEt �!1, then by (7), we conclude that I(zk)!�1 since �þ 142.(b) If ukk kEt� C, then vkk kEt!1:

Since F(u)� 0 and G(v)� 0, we estimateZ�

Fðuk þ vkÞdx � 0 � �C�

ukk kEtþ vkk kEtþ1�

and Z�

G At�sðuk � vkÞ� �

dx � 0 � �C ukk kEtþ vkk kEtþ1ð Þ,


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which implies that

IðzkÞ � ukk k2Et� vkk k

2Et þ 2C ukk kEtþ vkk kEtþ1ð Þ:

Then, I(zk)!�1 when kzkkE!þ1.

(A4) Now, we prove the (PS*) condition. Let (zn), zn2En, n2N, be a sequence

such that

InðznÞ�� C ð8Þ

and

I 0nðznÞ, � �� "n �k kE, for all �2En, and "n! 0: ð9Þ

Without loss of generality, and following the spirit of [4,11], we base our proof on the

fact that zn2E.

We first show that (zn) is uniformly bounded in E.

By (8) and (9), we have for zn¼ (un, vn)2E,

IðznÞ �1

2I0ðznÞ, zn �

¼

Z�

1

2f ðunÞun � FðunÞ

� dxþ

Z�

1

2gðvnÞvn � GðvnÞ

� dx

� Cþ "n znk kE:

By the expressions of F(u),G(v) and their derivatives, we can see that there exist

constants R40 and 25�5�þ 1 such that

05 �FðuÞ � f ðuÞu, and 05 �GðvÞ � gðvÞv for uj j, vj j � R:

Consequently,

1

2�

1

�

� Z�

�f ðunÞun�� þ gðvnÞvn

�� dx� � Cþ "n znk kE:

Hence, Z�

f ðunÞun�� dx � Cþ "n znk kE ð10Þ

and Z�

gðvnÞvn�� dx � Cþ "n znk kE: ð11Þ

Also, for zþn 2Eþ we obtain

zþn�� 2

E�"n zþn

�� E�

Z�

fðunÞuþn

�� dxþZ�

gðvnÞvþn

�� dx�

Z�

fðunÞ�� qþ1q dx� � q

qþ1

uþn��

qþ1þ

Z�

gðvnÞ�� pþ1p dx� � p

pþ1

vþn��

pþ1

�C

Z�

fðunÞ�� f ðunÞ�� 1qdx� � q

qþ1

uþn��

Etþ

Z�

gðvnÞ�� gðvnÞ�� 1pdx� � p

pþ1

vþn��

Es

!

�C 1þ

Z�

fðunÞun�� dx� q

qþ1

þ

Z�

gðvnÞvn�� dx� p

pþ1

!zþn��

E:


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Thus,

zþn��

E� C 1þ

Z�

f ðunÞun�� q

qþ1

dxþ

Z�

gðvnÞvn�� p

pþ1

dx

!:

Using the relations (10) and (11), we get

zþn��

E� C 1þ

hCþ "n znk kE

i qqþ1

þ

hCþ "n znk kE

i ppþ1

� :

Similarly, we have

z�n��

E� C 1þ

hCþ "n znk kE

i qqþ1

þ

hCþ "n znk kE

i ppþ1

� :

Since zn ¼ zþn þ z�n , we finally get for a new constant

znk kE� C 1þhCþ "n znk kE

i qqþ1

þ

hCþ "n znk kE

i ppþ1

� :

Thus, (zn) is uniformly bounded in E.Now we prove that (zn) converges strongly in E.Since (zn)¼ (un, vn) is bounded in E¼Et

�Es, there exists a subsequence denoted

again by (un, vn) which converges weakly to (u, v) in Et�Es and strongly in

Lqþ1(�)�Lpþ1(�).Since At : Et

!L2(�) and A�s : L2(�)!Es are continuous isomorphisms, we get

At(un� u)* 0 in L2(�), At�s(un� u)* 0 and At�s(un� u)! 0 strongly in Lpþ1(�).

Moreover, jvjp is strongly convergent in Lpþ1p :

Choosing �¼ (0,At�s(un� u))2Et�Es in (9) we thus conclude thatZ

�

�Atun�� 2�AtunA

tu�dx

�� Z�

�vþ v�j jp At�sðun � uÞ

�� þ vp� At�sðun � uÞ�� dx

þ "n At�sðun � uÞ��

Et

� C vpj jpþ1pAt�sðun � uÞ��

pþ1þ At�sðun � uÞ��

p1

� �þ "n At�sðun � uÞ

�� Et :

Observe that the right-hand side of the above inequality converges to 0, thusZ�

Atun�� 2dx�! Z

�

Atu�� 2dx:

Similarly, we prove that the sequence ( vn) converges strongly in Es.This implies that the (PS*) condition is satisfied, and thus, the conditions of

Theorem 3.2 are satisfied. By the regularity Theorem, the system ðeS�Þ has a strong

solution. Hence, the system (S�) admits a second solution (u�þ u, v�þ v). g

4. Non existence result

In this section we prove Theorem 1.3.Let �*¼ supf�02R

þj(S�) has at least one solution for �2 (0, �0)}. We want

to prove that there exists a positive number �* such that (S�) has no solution


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for �4�*. If the system (S�) has a solution (u�, v�) with respect to the parameter �,then we have

�

Z�

f ðxÞ�1dx ¼

Z�

h�1ðu� þ v�Þ � ðu

q� þ v

p�Þ

i�1dx:

By evaluating the minimum of ðuq� þ v

p�Þ � �1ðu� þ v�Þ on R

þ�R

þ, we can:

ðuq� þ v

p�Þ � �1ðu� þ v�Þ � �1 a

1

q� 1

� þ b

1

p� 1

� � ,

where a ¼ �1q

� � 1q�1

and b ¼ �1p

� � 1p�1

:

Using the fact that the problem (1) has a nonnegative solution uf, we have

�

Z�

Duf�1dx ¼ �1

Z�

uf �1dx ¼

Z�

f ðxÞ�1dx:

Thus Z�

f ðxÞ�1dx4 0: ð12Þ

Using (12), we have

� � �� :¼ �1 a�1�

1

q

�þ b

�1�

1

p

�� Z�

�1dxZ�

f ðxÞ�1dx

:

This immediately implies that there is a number �* such that (S�) has no solution

for �4�*. g

References

[1] N.P. Cac, J.A. Gatica, and Y. Li, Positive solutions to semilinear problems with coefficient

that changes sign, Nonlinear Anal. 37 (1999), pp. 501–510.

[2] D.G. De Figuerido and P. Felmer, On superquadratic elliptic systems, Trans. Amer. Math.

Soc. 343 (1994), pp. 99–116.

[3] J. Hulshof and R.C.A.M. Van der Vorst, Differential systems with strongly indefinite

variational structure, J. Funct. Anal. 114 (1993), pp. 32–58.

[4] D.G. De Figuerido and B. Ruf, Elliptic systems with nonlinearities of arbitrary growth,

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[5] V. Benci and P.H. Rabinowitz, Critical point theorems for indefinite functional, Invent.

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[9] J. Hernandez, Qualitative methods for non-linear diffusion equations, in Nonlinear DiffusionProblems, Vol. 1224, A. Dobb and B. Eckmann, eds., Lecture Notes in Mathematics,Springer Verlag, Berlin, 1986, pp. 47–118.

[10] D.G. De Figueiredo, Semilinear elliptic systems, in Nonlinear Functional Analysis and

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