Solutions of the Dirichlet Schrödinger problems with continuous data admitting arbitrary growth property in the boundary

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R E S E A R C H

Open Access

Solutions of the Dirichlet-Schrödinger

problems with continuous data admitting

arbitrary growth property in the boundary

Jianjie Wang

1*

_{, Jun Pu}

2

_{and Ahmed Zama}

3 Full list of author information is available at the end of the article

Abstract

By using the modiﬁed Green-Schrödinger function, we consider the Dirichlet problem with respect to the stationary Schrödinger operator with continuous data having an arbitrary growth in the boundary of the cone. As an application of the modiﬁed Poisson-Schrödinger integral, the unique solution of it is also constructed.

Keywords: modiﬁed Green-Schrödinger potential; modiﬁed Poisson-Schrödinger integral; Dirichlet-Schrödinger problem

1 Introduction and main theorem

We denote then-dimensional Euclidean space byRn_{, where}_n_≥_{. The sets}_∂E _and_E

denote the boundary and the closure of a setEinRn_{. Let}_|_V_–_W_|_{denote the Euclidean}

distance of two pointsVandWinRn, respectively. Especially,|V|denotes the distance of two pointsVandOinRn, whereOis the origin ofRn.

+ denote the unit sphere and the upper half unit sphere inRn, respectively.

The surface area πn/{(n/)}–_of_Sn–_{is denoted by}_w

n. Let⊂Sn–,anddenote

a point (,) and the set{; (,)∈}, respectively. For two sets⊂R+and⊂Sn–,

we denote

×=(τ,)∈Rn;τ∈, (,)∈,

©2016 Wang et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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whereR+is the set of all positive real numbers.

For the set⊂Sn–_{, a cone}_H

n() denote the setR+×inRn. For the setE⊂R,Cn(;I)

andSn(;I) denote the setsE×andE×∂, respectively, whereRis the set of all real

numbers. Especially,Sn() denotes the setSn(;R+).

LetAadenote the class of nonnegative radial potentialsa(V),i.e.≤a(V) =a(τ),V=

(τ,)∈Hn(), such thata∈Lb_loc(Hn()) with someb>n/ ifn≥ and withb=  ifn= 

orn= .

This article is devoted to the stationary Schrödinger equation

SSE_au(V) = – nu(V) +a(V)u(V) = ,

forV ∈Cn(), where nis the Laplace operator anda∈Aa. These solutions are called

harmonic functions with respect toSSEa. In the casea=  we remark that they are

har-monic functions. Under these assumptions the operatorSSEacan be extended in the usual

way from the spaceC∞_ (Hn()) to an essentially self-adjoint operator onL(Hn()) (see

[]). We will denote itSSE_aas well. This last one also has a Green-Schrödinger function

G(;a)(V,W). HereG(;a)(V,W) is positive onHn() and its inner normal derivative

∂G(;a)(V,W)/∂nW≥. We denote this derivative byPI(;a)(V,W), which is called the

Poisson-Schrödinger kernel with respect toHn().

Let be the spherical part of the Laplace operator on⊂Sn–_and_λ

j(j= , ,  . . . ,

 <λ<λ≤λ≤. . .) be the eigenvalues of the eigenvalue problem for on(see,e.g.,

[], p.)

_ϕ₍_{) +}_λϕ₍_{) = } _in_,

ϕ() =  on∂.

The corresponding eigenfunctions are denoted byϕjv(≤v≤vj), wherevjis the

multi-plicity ofλj. We setλ= , norm the eigenfunctions inL(), andϕ=ϕ> .

We wish to ensure the existence ofλj, wherej= , ,  . . . . We put a rather strong

as-sumption on: ifn≥, thenis aC,α_{-domain ( <}_α_{< ) on S}n–_{surrounded by a ﬁnite}

number of mutually disjoint closed hypersurfaces (e.g.see [], pp.- for the deﬁnition of aC,α_-domain).

Given a continuous functionf onSn(), we say thathis a solution of the

Dirichlet-Schrödinger problem inHn() withf, ifhis a harmonic function with respect toSSEain

Hn() and

lim

V→W∈Sn(),V∈Hn()h(V) =f(W).

The solutions of the equation –(τ) – n– 

τ

₍_τ_{) +}λj

τ+a(τ)

(τ) = ,  <τ<∞, (.) are denoted byPj(τ) (j= , , , . . .) andQj(τ) (j= , , , . . .), respectively, for the increasing

and non-increasing cases, asτ →+∞, which is normalized under the conditionPj() =

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Qj() =  (see [], Chap. ). In the sequel, we shall writeP andQinstead ofP andQ,

respectively, for the sake of brevity.

We shall also consider the classBa, consisting of the potentialsa∈Aasuch that there

exists a ﬁnite limitlimτ→∞τa(τ) =k∈[,∞), moreover,τ–|τa(τ) –k| ∈L(,∞). Ifa∈ Ba, then the generalized harmonic functions are continuous (see []).

In the rest of this paper, we assume thata∈Baand we shall suppress the explicit notation

of this assumption for simplicity. Denote

ζ_j±_,_k= –n± (n– )

_{+ (}_k₊_λ

j)

 forj= , , ,  . . . .

It is well known (see []) that in the case under consideration the solutions to equation (.) have the asymptotics wheredanddare some positive constants.

The Green-Schrödinger function G(;a)(V,W) (see [], Chap. ) has the following

The modiﬁed Green-Schrödinger function can be deﬁned as follows (see [], Chap. ):

G(;a,m)(V,W) =G(;a)(V,W) –K(;a,m)(V,W)

for two pointsV= (τ,),Q= (ι,ϒ)∈Hn(), then the modiﬁed Poisson-Schrödinger case

on cones can be deﬁned by

PI(;a,m)(V,W) =∂G(;a,m)(V,W)

∂nW

accordingly, which has the following growth estimates (see []):

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for any V = (τ,)∈Hn() andW = (ι,ϒ)∈Sn() satisfyingτ ≤sι ( <s< ), where

M(n,m,s) is a constant dependent ofn,m, ands. We remark that

PI(;a, )(V,W) =PI(;a)(V,W).

In this paper, we shall use the following modiﬁed Poisson-Schrödinger integrals (see []):

PIa

(m,f)(V) =

Sn()

PI(;a,m)(V,W)f(W)dσW,

wheref(W) is a continuous function on∂Hn() anddσW is the surface area element on

Sn().

For more applications of modiﬁed Green-Schrödinger potentials and modiﬁed Poisson-Schrödinger integrals, we refer the reader to the papers (see [, ]).

Recently, Huang and Ychussie (see []) gave the solutions of the Dirichlet-Schrödinger problem with continuous data having slow growth in the boundary.

Theorem A If f is a continuous function on∂Hn()satisfying

Sn()

|f(ι,ϒ)|  +Pm+(ι)ιn–

dσW<∞, (.)

then the modiﬁed Poisson-Schrödinger integral PIa(m,f) is a solution of the Dirichlet-Schrödinger problem in Hn()with f satisﬁes

lim τ→∞,V=(τ,)∈Hn()τ

–ζ_m+_+,_k

ϕ_n–()PIa(m,f)(V) = .

It is natural to ask if the continuous functionfsatisfying (.) can be replaced by contin-uous data having an arbitrary growth property in the boundary. In this paper, we shall give an aﬃrmative answer to this question. To do this, we also construct a modiﬁed Poisson-Schrödinger kernel. Letφ(l) be a positive function ofl≥ satisfying

P()φ() = . Denote the set

l≥; –ζ_j+_,_klog =logln–φ(l)

by π(φ,j). Then ∈π(φ,j). When there is an integerN such that π(φ,N)=and

π(φ,N+ ) =, denote

J(φ) ={j; ≤j≤N}

of integers. Otherwise, denote the set of all positive integers byJ(φ). Letl(j) =l(φ,j) be the minimum elementslinπ(φ,j) for eachj∈J(φ). In the former case, we putl(N+ ) =

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∞. Thenl() = . The kernel functionK(;a,φ)(V,W) is deﬁned by

K(;a,φ)(V,W) =

 if  <t< ,

K(;a,j)(V,W) ifl(j)≤t<l(j+ ) andj∈J(φ), whereV∈Hn() andW= (ι,ϒ)∈Sn().

The new modiﬁed Poisson-Schrödinger kernelPI(;a,φ)(V,W) is deﬁned by

PI(;a,φ)(V,W) =PI(;a)(V,W) –K(;a,φ)(V,W), whereV∈Hn() andW∈Sn().

As an application of modiﬁed Poisson-Schrödinger kernel PI(;a,φ)(V,W), we have the following.

Theorem Let g(V)be a continuous function on Sn().Then there is a positive continuous

functionφg(l)of l≥depending on g such that

PIa

(φg,g)(V) =

Sn()PI

(;a,φg)(V,W)g(W)dσW

is a solution of the Dirichlet-Schrödinger problem in Hn()with g.

2 Main lemmas

Lemma  Letφ(l)be a positive continuous function of l≥satisfying P()φ() = .

Then

_PI(;a)(V,W) –K(;a,φ)≤Mφ(l)

for any V = (τ,)∈Hn()and any W= (ι,ϒ)∈Sn()satisfying

ι>max{, r}. (.)

Proof We can choose two pointsV = (τ,)∈Hn() andW = (ι,ϒ)∈Sn(), satisfying

(.). Moreover, we also can choose an integerj=j(V,W)∈J(ϒ) such that

l(j– )≤ι<l(j). (.) Then

K(;a,φ)(V,W) =K(;a,j– )(V,W). Hence we have from (.), (.), and (.)

_PI(;a)(V,W) –K(;a,φ)(V,W)≤M–ζki+ ≤_Mφ₍_l_),

which is the conclusion.

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Lemma (see []) Let g(V)be a continuous function on Sn()andV(V,W)be a locally

integrable function on Sn()for any ﬁxed V∈Hn(),where W∈Sn().Deﬁne

W(V,W) =PI(;a)(V,W) –V(V,W)

for any V ∈Hn()and any W∈Sn().

Suppose that the following two conditions are satisﬁed:

(I) For anyQ∈Sn()and any> ,there exists a neighborhoodB(Q)ofQsuch that

for any positive real numberR. Then

Then we see from Lemma  that

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Obviously, we have

Sn(;(,R))

_PI(;a,φg)(V,W)g(W)dσW<∞,

which gives

Sn()

_PI(;a,φg)(V,W)g(W)dσW<∞.

To see thatPIa(φg,g)(V) is a harmonic function inHn(), we remark thatPIa(φg,g)(V)

satisﬁes the locally mean-valued property by Fubini’s theorem. Finally we shall show that

lim

V∈Hn(),V→W

PIa

(φg,g)(V) =g

W

for anyW= (ι,ϒ)∈∂Hn(). Setting

V(V,W) =K(;a,φg)(V,W)

in Lemma , which is locally integrable onSn() for any ﬁxedV∈Hn(). Then we apply

Lemma  tog(V) and –g(V).

For any>  and a positive numberδ, by (.) we can choose a numberR(>max{, (ι+δ)}) such that (.) holds, whereV∈Hn()∩B(W,δ).

Since

lim

→ϕi() =  (i= , ,  . . .)

asV= (τ,)→W= (ι,)∈Sn(), we have

lim

V∈Hn(),V→ϒ

K(;a,φg)(V,W) = ,

whereW∈Sn() andW∈Sn(). Then (.) holds.

Thus we complete the proof of the theorem.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the ﬁnal manuscript.

Author details

1_{College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030031, China.}2_{Center for}

Finance and Accounting Research, University of International Business and Economics, Beijing, 100029, China. 3_{Department of Computer Engineering, College of Engineering, University of Mosul, Mosul, Iraq.}

Acknowledgements

We wish to express our genuine thanks to the anonymous referees for careful reading and excellent comments on this manuscript.

Received: 15 October 2015 Accepted: 11 January 2016

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