Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 DOI 10.1186/s13663-015-0342-1

R E S E A R C H Open Access

Sharp geometrical properties of a-rarefiedsets via fixed point index for the Schrödingeroperator equationsZhiqiang Li1 and Beatriz Ychussie2*

*Correspondence:ychussie.b@gmail.com2Mathematics Institute, RoskildeUniversity, Roskilde, 4000, DenmarkFull list of author information isavailable at the end of the article

AbstractIn this paper, we use the theory of fixed point index for the Schrödinger operatorequations to obtain a geometrical property of a-rarefied sets at infinity on cones.Meanwhile, we give an example to show that the reverse of this property is not true.

Keywords: Schrödinger operator equations; rarefied set; Poisson-Sch integral;Green-Sch potential

1 Introduction and main theoremLet R and R+ be the set of all real numbers and the set of all positive real numbers, re-spectively. We denote by Rn (n ≥ ) the n-dimensional Euclidean space. A point in Rn isdenoted by P = (X, xn), X = (x, x, . . . , xn–). The Euclidean distance between two points Pand Q in Rn is denoted by |P – Q|. Also |P – O| with the origin O of Rn is simply denotedby |P|. The boundary and the closure of a set S in Rn are denoted by ∂S and S, respec-tively. For P ∈ Rn and r > , let B(P, r) denote the open ball with center at P and radius rin Rn.

We introduce a system of spherical coordinates (r,�), � = (θ, θ, . . . , θn–), in Rn whichare related to Cartesian coordinates (x, x, . . . , xn–, xn) by xn = r cos θ.

Let D be an arbitrary domain in Rn and Aa denote the class of nonnegative radial po-tentials a(P), i.e. ≤ a(P) = a(r), P = (r,�) ∈ D, such that a ∈ Lb

loc(D) with some b > n/ ifn ≥ and with b = if n = or n = .

If a ∈ Aa, then the Schrödinger operator

Scha = –� + a(P)I = ,

where � is the Laplace operator and I is the identical operator, can be extended in theusual way from the space C∞

(D) to an essentially self-adjoint operator on L(D) (see [],Chapter ). We will denote it by Scha as well. This last one has a Green-Sch functionGa

D(P, Q). Here GaD(P, Q) is positive on D and its inner normal derivative ∂Ga

D(P, Q)/∂nQ ≥, where ∂/∂nQ denotes the differentiation at Q along the inward normal into D.

We call a function u �≡ –∞ that is upper semi-continuous in D a subfunction with re-spect to the Schrödinger operator Scha if its values belong to the interval [–∞,∞) and at

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Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 Page 2 of 9

each point P ∈ D with < r < r(P) the generalized mean-value inequality (see [])

u(P) ≤∫

∂B(P,r)u(Q)

∂GaB(P,r)(P, Q)∂nQ

dσ (Q)

is satisfied, where GaB(P,r)(P, Q) is the Green-Sch function of Scha in B(P, r) and dσ (Q) is a

surface measure on the sphere ∂B(P, r).If –u is a subfunction, then we call u a superfunction. If a function u is both subfunction

and superfunction, it is, clearly, continuous and is called a generalized harmonic function(with respect to the Schrödinger operator Scha).

The unit sphere and the upper half unit sphere in Rn are denoted by Sn– and Sn–+ ,

respectively. For simplicity, a point (,�) on Sn– and the set {�; (,�) ∈ �} for a set�, � ⊂ Sn–, are often identified with � and �, respectively. For two sets � ⊂ R+ and� ⊂ Sn–, the set {(r,�) ∈ Rn; r ∈ �, (,�) ∈ �} in Rn is simply denoted by � × �. ByCn(�), we denote the set R+ × � in Rn with the domain � on Sn–. We call it a cone. Wedenote the set I × � with an interval on R by Cn(�; I).

We shall say that a set H ⊂ Cn(�) has a covering {rj, Rj} if there exists a sequence of balls{Bj} with centers in Cn(�) such that H ⊂ ⋃∞

j= Bj, where rj is the radius of Bj and Rj is thedistance from the origin to the center of Bj. For positive functions h and h, we say thath � h if h ≤ Mh for some constant M > . If h � h and h � h, we say that h ≈ h.

From now on, we always assume D = Cn(�). For the sake of brevity, we shall writeGa

�(P, Q) instead of GaCn(�)(P, Q). Throughout this paper, let c denote various positive con-

stants, because we do not need to specify them. Moreover, ε appearing in the expressionin the following all sections will be a sufficiently small positive number.

Let � be a domain on Sn– with smooth boundary. Consider the Dirichlet problem

(n + λ)ϕ = on �,

ϕ = on ∂�,

where n is the spherical part of the Laplace operator �n:

�n =n –

r∂

∂r+

∂

∂r +n

r .

We denote the least positive eigenvalue of this boundary value problem by λ and the nor-malized positive eigenfunction corresponding to λ by ϕ(�). In order to ensure the exis-tence of λ and a smooth ϕ(�). We put a rather strong assumption on �: if n ≥ , then �

is a C,α-domain ( < α < ) on Sn– surrounded by a finite number of mutually disjointclosed hypersurfaces.

Solutions of an ordinary differential equation

–Q′′(r) –n –

rQ′(r) +

(λ

r + a(r))

Q(r) = , < r < ∞. (.)

It is well known (see, for example, []) that if the potential a ∈ Aa, then (.) has a fun-damental system of positive solutions {V , W } such that V and W are increasing and de-creasing, respectively (see [–]).

Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 Page 3 of 9

We will also consider the class Ba, consisting of the potentials a ∈ Aa such that thereexists the finite limit limr→∞ ra(r) = k ∈ [,∞), and moreover, r–|ra(r) – k| ∈ L(,∞). Ifa ∈ Ba, then the (sub)superfunctions are continuous (see []).

In the rest of paper, we assume that a ∈ Ba and we shall suppress this assumption forsimplicity.

Denote

ι±k = – n ± √

(n – ) + (k + λ)

,

then the solutions to (.) have the asymptotics (see [])

V (r) ≈ rι+k , W (r) ≈ rι–k , as r → ∞. (.)

Let ν be any positive measure on cones such that the Green-Sch potential

Ga�ν(P) =

∫Cn(�)

Ga�(P, Q) dν(Q) �≡ +∞

for any P ∈ Cn(�). Then the positive measure ν ′ on Rn is defined by

dν ′(Q) =

{W (t)ϕ(�) dν(Q), Q = (t,�) ∈ Cn(�; (, +∞)),, Q ∈ Rn – Cn(�; (, +∞)).

The Poisson-Sch integral PIa�μ(P) �≡ +∞ (P ∈ Cn(�)) of μ on cones is defined as follows:

PIa�μ(P) =

cn

∫Sn(�)

PIa�(P, Q) dμ(Q),

where

PIa�(P, Q) =

∂Ga�(P, Q)∂nQ

, cn =

{π , n = ,(n – )sn, n ≥ ,

μ is a positive measure on ∂Cn(�) and ∂/∂nQ denotes the differentiation at Q along theinward normal into cones. Then the positive measure μ′ on Rn is defined by

dμ′(Q) =

{t–W (t) ∂ϕ(�)

∂n�dμ(Q), Q = (t,�) ∈ Sn(�; (, +∞)),

, Q ∈ Rn – Sn(�; (, +∞)).

Remark We remark that the total masses of μ′ and ν ′ are finite (see [], Lemma and [],Lemma ).

Let ≤ α ≤ n and λ be any positive measure on Rn having finite total mass. For eachP = (r,�) ∈ Rn – {O}, the maximal function M(P;λ,α) with respect to Scha is defined by

M(P;λ,α) = sup<ρ< r

λ(B(P,ρ)

)V (ρ)W (ρ)ρα–.

Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 Page 4 of 9

The set

{P = (r,�) ∈ Rn – {O}; M(P;λ,α)V –(r)W –(r)r–α > ε

}

is denoted by E(ε;λ,α).The following Theorems A and B give a way to estimate the Green-Sch potential and

the Poisson-Sch integrals with measures on Cn(�) and Sn(�), respectively.

Theorem A Let ν be a positive measure on Cn(�) such that Ga�ν(P) �≡ +∞ (P = (r,�) ∈

Cn(�)) holds. Then for a sufficiently large L we have

{P ∈ Cn

(�; (L, +∞)

); Ga

�ν(P) ≥ V (r)} ⊂ E

(ε;μ′,

).

Theorem B Let μ be a positive measure on Sn(�) such that PIa�μ(P) �≡ +∞ (P = (r,�) ∈

Cn(�)). Then for a sufficiently large L we have

{P ∈ Cn

(�; (L, +∞)

); PIa

�μ(P) ≥ V (r)} ⊂ E

(ε;μ′,

).

It is known that the Martin boundary of Cn(�) is the set ∂Cn(�) ∪ {∞}, each of whichis a minimal Martin boundary point. For P ∈ Cn(�) and Q ∈ ∂Cn(�) ∪ {∞}, the Martinkernel can be defined by Ma

�(P, Q). If the reference point P is chosen suitably, then we have

Ma�(P,∞) = V (r)ϕ(�) and Ma

�(P, O) = cW (r)ϕ(�)

for any P = (r,�) ∈ Cn(�).In [, ], Xue and Zhao-Yamada introduce the notations of a-thin (with respect to

the Schrödinger operator Scha) at a point and a-rarefied sets at infinity (with respectto the Schrödinger operator Scha), which generalized the earlier notations obtained byMiyamoto, Hoshida, Brelot (see [–]).

Definition (see []) A set H in Rn is said to be a-thin at a point Q if there is a fineneighborhood E of Q which does not intersect H\{Q}. Otherwise H is said to be not a-thinat Q on cones.

Definition (see []) A subset H of Cn(�) is said to be a-rarefied at infinity on cones, ifthere exists a positive superfunction v(P) on cones such that

infP∈Cn(�)

v(P)Ma

�(P,∞)≡ (.)

and

H ⊂ {P = (r,�) ∈ Cn(�); v(P) ≥ V (r)

}. (.)

Let H be a bounded subset of Cn(�). Then R̂HMa

�(·,∞) is bounded on cones and the greatestgeneralized harmonic minorant of R̂H

Ma�(·,∞) is zero. We see from the Riesz decomposition

Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 Page 5 of 9

theorem (see [], Theorem ) that there exists a unique positive measure λaH on cones such

that (see [], p.)

R̂HMa

�(·,∞)(P) = Ga�λa

H (P)

for any P ∈ Cn(�) and λaH is concentrated on IH , where

IH ={

P ∈ Cn(�); H is not a-thin at P}

.

We denote the total mass λaH(Cn(�)) of λa

H by λa�(H).

Recently, GX Xue (see [], Theorem .) gave a criterion for a subset H of Cn(�) to bea-rarefied set at infinity.

Theorem C A subset H of Cn(�) is a-rarefied at infinity on cones if and only if

∞∑j=

W(j)λa

Hj

(Cn(�)

)< ∞,

where Hj = H ∩ Cn(�; [j, j+)) and j = , , , . . . .

Our aim in this paper is to characterize the geometrical property of a-rarefied sets atinfinity.

Theorem If a subset H of Cn(�) is a-rarefied at infinity on cones, then H has a covering{rj, Rj} (j = , , , . . .) satisfying

∞∑j=

(rj

Rj

)V (Rj)V (rj)

W (Rj)W (rj)

< ∞. (.)

Next, we immediately have the following result from Theorem .

Corollary Let v(P) be positive superfunction on cones. Then v(P)V –(r) uniformly con-verges to c∞(v, a)ϕ(�) as r → ∞ outside a set which has a covering {rj, Rj} (j = , , , . . .)satisfying (.), where

c∞(v, a) = infP∈Cn(�)

v(P)Ma

�(P,∞).

Finally, we prove the following result.

Theorem If a subset H of Cn(�) has a covering {rj, Rj} (j = , , , . . .) satisfying (.), thenit is possible that H is not a-rarefied at infinity on cones.

2 Main lemmasLemma Let λ be any positive measure on Rn having finite total mass. Then E(ε;λ, ) hasa covering {rj, Rj} (j = , , . . .) satisfying

∞∑j=

(rj

Rj

)V (Rj)W (Rj)V (rj)W (rj)

< ∞.

Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 Page 6 of 9

Proof Set

Ej(ε;λ, ) ={

P = (r,�) ∈ E(ε;λ, ) : j ≤ r < j+} (j = , , , . . .).

If P = (r,�) ∈ Ej(ε;λ, ), then there exists a positive number ρ(P) such that

(ρ(P)

r

)V (r)W (R)

V (ρ(P))W (ρ(P))≈

(ρ(P)

r

)n–

≤ λ(B(P,ρ(P)))ε

.

Since Ej(ε;λ, ) can be covered by the union of a family of balls {B(Pj,i,ρj,i) : Pj,i ∈Ek(ε;λ, )} (ρj,i = ρ(Pj,i)). By the Vitali lemma (see []), there exists j ⊂ Ej(ε;λ, ),which is at most countable, such that {B(Pj,i,ρj,i) : Pj,i ∈ j} are disjoint and Ej(ε;λ, ) ⊂⋃

Pj,i∈jB(Pj,i, ρj,i). So

∞⋃j=

Ej(ε;λ, ) ⊂∞⋃j=

⋃Pj,i∈j

B(Pj,i, ρj,i).

On the other hand, note that

⋃Pj,i∈j

B(Pj,i,ρj,i) ⊂ {P = (r,�) : j– ≤ r < j+},

so that

∑Pj,i∈j

(ρj,i

|Pj,i|)

V (|Pj,i|)W (|Pj,i|)V (ρj,i)W (ρj,i)

≈∑

Pj,i∈j

(ρj,i

|Pj,i|)n–

≤ n–∑

Pj,i∈j

λ(B(Pj,i,ρj,i))ε

≤ n–

ελ(Cn

(�;

[j–, j+))).

Hence we obtain

∞∑j=

∑Pj,i∈j

(ρj,i

|Pj,i|)

V (|Pj,i|)W (|Pj,i|)V (ρj,i)W (ρj,i)

≈∞∑j=

∑Pj,i∈j

(ρj,i

|Pj,i|)n–

≤∞∑j=

λ(Cn(�; [j–, j+)))ε

≤ λ(Rn)ε

.

Since E(ε;λ, )∩{P = (r,�) ∈ Rn; r ≥ } =⋃∞

j= Ej(ε;λ, ). Then E(ε;λ, ) is finally coveredby a sequence of balls {B(Pj,i,ρj,i), B(P, )} (j = , , . . . ; i = , , . . .) satisfying

∑j,i

(ρj,i

|Pj,i|)

V (|Pj,i|)W (|Pj,i|)V (ρj,i)W (ρj,i)

≈∑

j,i

(ρj,i

|Pj,i|)n–

≤ λ(Rn)ε

+ n–α < +∞,

where B(P, ) (P = (, , . . . , ) ∈ Rn) is the ball which covers {P = (r,�) ∈ Rn; r < }. �

Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 Page 7 of 9

3 Proof of Theorem 1Since H is a-rarefied at infinity on cones, by Definition there exists a positive superfunc-tion v(P) on cones such that (.) and (.) hold.

For this v(P) there exists a unique positive measure μ′′ on Sn(�) and a unique positivemeasure ν ′′ on cones such that (see [], Theorem )

v(P) = c(v, a)Ma�(P, O) + Ga

�ν ′′(P) + PIa�μ′′(P), (.)

where

c(v, a) = infP∈Cn(�)

v(P)Ma

�(P, O).

Let us denote

H ={

P = (r,�) ∈ Cn(�); c(v, a)Ma�(P, O) ≥ V (r)

},

H ={

P = (r,�) ∈ Cn(�); Ga�ν ′′(P) ≥ V (r)

}

and

H ={

P = (r,�) ∈ Cn(�); PIa�μ′′(P) ≥ V (r)

},

respectively.Then we see from (.) that

H ⊂ H ∪ H ∪ H. (.)

For each Hi (i = , , ), we know that it has a covering. It is evident from the boundednessof H that H has a covering {r, R} satisfying

r

R< +∞. (.)

When we apply Theorems A and B with the measures μ and ν defined by μ = μ′′ andν = ν ′′, respectively, we can find two positive constants L and ε such that

H ∩ Cn(�; (L, +∞)

) ⊂ E(ε;μ′,

)

and

H ∩ Cn(�; (L, +∞)

) ⊂ E(ε;ν ′,

),

respectively.By Lemma , these sets E(ε;μ′, ) and E(ε;ν ′, ) have coverings {r()

j , R()j } (j = , , . . .) and

{r()j , R()

j } (j = , , . . .) satisfying

∞∑j=

( r()j

R()j

)V (R()j )W (R()

j )

V (r()j )W (r()

j )< +∞ (.)

Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 Page 8 of 9

and

∞∑j=

( r()j

R()j

)V (R()j )W (R()

j )

V (r()j )W (r()

j )< +∞, (.)

respectively.Then H and H also have coverings {r()

j , R()j } (j = , , . . .) and {r()

j , R()j } (j = , , . . .)

satisfying (.) and (.), respectively.Thus by rearranging coverings {r, R}, {r()

j , R()j } (j = , , . . .) and {r()

j , R()j } (j = , , . . .),

we know that the set H has a covering {rj, Rj} (j = , , , . . .) from (.) and satisfies (.)from (.), (.), and (.).

Thus we complete the proof of Theorem .

4 Proof of Theorem 2Put

rj = · j– · j

–n and Rj = · j– (j = , , , . . .).

A covering {rj, Rj} satisfies

∞∑j=

(rj

Rj

)V (Rj)V (rj)

W (Rj)W (rj)

≤ c∞∑j=

(rj

Rj

)n–

= c∞∑j=

jn––n < +∞

from (.).Let Cn(�′) be a subset of Cn(�), i.e. �

′ ⊂ �. Suppose that this covering is so located:there is an integer j such that Bj ⊂ Cn(�′) and Rj > rj for j ≥ j.

Next we shall prove that the set H =⋃∞

j=j Bj is not a-rarefied at infinity on Cn(�). Sinceϕ(�) ≥ c for any � ∈ �′, we have Ma

�(P,∞) ≥ cV (Rj) for any P ∈ Bj, where j ≥ j. Hencewe have

R̂BjMa

�(·,∞)(P) ≥ cV (Rj) (.)

for any P ∈ Bj, where j ≥ j.Take a measure δ on cones, supp δ ⊂ Bj, δ(Bj) = such that

∫Cn(�)

|P – Q|–n dδ(P) ={Cap(Bj)

}– (.)

for any Q ∈ Bj, where Cap denotes the Newton capacity. Since

Ga�(P, Q) ≤ |P – Q|–n

for any P ∈ Cn(�) and Q ∈ Cn(�),

{Cap(Bj)

}–λa

Bj

(Cn(�)

)=

∫ (∫|P – Q|–n dδ(P)

)dλa

Bj(Q)

≥∫ (∫

Ga�(P, Q) dλa

Bj(Q)

)dδ(P)

Li and Ychussie Fixed Point Theory and Applications (2015) 2015:89 Page 9 of 9

=∫

R̂BjMa

�(·,∞) dδ(P)

≥ cV (Rj)δ(Bj) = cV (Rj)

from (.) and (.). Hence we have (see [], p.)

λaBj

(Cn(�)

) ≥ c Cap(Bj)V (Rj) ≥ crn–j V (Rj). (.)

If we observe λaHj

(Cn(�)) = λaBj

(Cn(�)), then we have by (.)

∞∑j=j

W(j)λa

Hj

(Cn(�)

) ≥ c∞∑

j=j

(rj

Rj

)n–

= c∞∑

j=j

j

= +∞,

from which it follows by Theorem C that H is not a-rarefied at infinity on cones.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsBoth authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Author details1Institute of Management Science and Engineering, Henan University, Kaifeng, 475001, China. 2Mathematics Institute,Roskilde University, Roskilde, 4000, Denmark.

AcknowledgementsThis work was completed while the second author was visiting the Department of Mathematical Sciences at theColumbia University, and he is grateful for the kind hospitality of the Department. This work was partially supported byNSF Grant DMS-0913205.

Received: 23 January 2015 Accepted: 28 May 2015

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