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

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.


Introduction and main theorem
We denote the n-dimensional Euclidean space by R n , where n ≥ . The sets ∂E and E denote the boundary and the closure of a set E in R n . Let |V -W | denote the Euclidean distance of two points V and W in R n , respectively. Especially, |V| denotes the distance of two points V and O in R n , where O is the origin of R n . We introduce a system of spherical coordinates (τ , ), = (λ  , λ  , . . . , λ n- ), in R n which are related to the Cartesian coordinates (y  , y  , . . . , y n- , y n ) by y  = τ n- j= sin λ j (n ≥ ), y n = τ cos λ  , and if n ≥ , then where  ≤ τ < +∞, -  π ≤ λ n- <   π , and if n ≥ , then  ≤ λ j ≤ π ( ≤ j ≤ n -). Let B(V , τ ) denote the open ball with center at V and radius r in R n , where τ > . Let S n- and S n- + denote the unit sphere and the upper half unit sphere in R n , respectively. The surface area π n/ { (n/)} - of S n- is denoted by w n . Let ⊂ S n- , and denote a point (, ) and the set { ; (, ) ∈ }, respectively. For two sets ⊂ R + and ⊂ S n- , we denote × = (τ , ) ∈ R n ; 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, provided 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.

R E T R
where R + is the set of all positive real numbers.
For the set ⊂ S n- , a cone H n ( ) denote the set R + × in R n . For the set E ⊂ R, C n ( ; I) and S n ( ; I) denote the sets E × and E × ∂ , respectively, where R is the set of all real numbers. Especially, S n ( ) denotes the set S n ( ; R + ).
Let A a denote the class of nonnegative radial potentials a(V ), i.e.  ≤ a(V ) = a(τ ), V = (τ , ) ∈ H n ( ), such that a ∈ L b loc (H n ( )) with some b > n/ if n ≥  and with b =  if n =  or n = .
This article is devoted to the stationary Schrödinger equation for V ∈ C n ( ), where n is the Laplace operator and a ∈ A a . These solutions are called harmonic functions with respect to SSE a . In the case a =  we remark that they are harmonic functions. Under these assumptions the operator SSE a can be extended in the usual way from the space C ∞  (H n ( )) to an essentially self-adjoint operator on L  (H n ( )) (see []). We will denote it SSE a as well. This last one also has a Green-Schrödinger function G( ; a)(V , W ). Here G( ; a)(V , W ) is positive on H n ( ) and its inner normal derivative ∂G( ; a)(V , W )/∂n W ≥ . We denote this derivative by PI( ; a)(V , W ), which is called the Poisson-Schrödinger kernel with respect to H n ( ).
Let be the spherical part of the Laplace operator on ⊂ S n- and λ j (j = , ,  . . . ,  < λ  < λ  ≤ λ  ≤ . . .) be the eigenvalues of the eigenvalue problem for on (see, e.g., The corresponding eigenfunctions are denoted by ϕ jv ( ≤ v ≤ v j ), where v j is the multiplicity of λ j . We set λ  = , norm the eigenfunctions in L  ( ), and ϕ  = ϕ  > .
We wish to ensure the existence of λ j , where j = , ,  . . . . We put a rather strong assumption on : if n ≥ , then is a C ,α -domain ( < α < ) on S n- surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [], pp.- for the definition of a C ,α -domain).
Given a continuous function f on S n ( ), we say that h is a solution of the Dirichlet-Schrödinger problem in H n ( ) with f , if h is a harmonic function with respect to SSE a in H n ( ) and The solutions of the equation are denoted by P j (τ ) (j = , , , . . .) and Q j (τ ) (j = , , , . . .), respectively, for the increasing and non-increasing cases, as τ → +∞, which is normalized under the condition P j () =

R E T R
In the sequel, we shall write P and Q instead of P  and Q  , respectively, for the sake of brevity. We shall also consider the class B a , consisting of the potentials a ∈ A a such that there exists a finite limit then the generalized harmonic functions are continuous (see []).
In the rest of this paper, we assume that a ∈ B a and we shall suppress the explicit notation of this assumption for simplicity. Denote It is well known (see []) that in the case under consideration the solutions to equation (.) have the asymptotics where d  and d  are some positive constants. The Green-Schrödinger function G( ; a)(V , W ) (see [], Chap. ) has the following expansion: for a ∈ A a , where V = (τ , ), W = (ι, ϒ), τ = ι, and χ (s) = w(Q  (τ ), P  (τ ))| τ =s is their Wronskian. The series converges uniformly if either τ ≤ sι or τ ≤ sι ( < s < ).
For a nonnegative integer m and two points V = (τ , ), W = (ι, ϒ) ∈ H n ( ), we put It is natural to ask if the continuous function f satisfying (.) can be replaced by continuous 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. Let φ(l) be a positive function of l ≥  satisfying Denote the set l ≥ ; -ζ + j,k log  = log l n- φ(l) by π (φ, j). Then  ∈ π (φ, j). When there is an integer N such that π (φ, N) = and π (φ, N + ) = , denote of integers. Otherwise, denote the set of all positive integers by J (φ). Let l(j) = l (φ, j) be the minimum elements l in π (φ, j) for each j ∈ J (φ). In the former case, we put l(N + ) =
The new modified Poisson-Schrödinger kernel PI( ; a, φ)(V , W ) is defined by where V ∈ H n ( ) and W ∈ S n ( ).
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 S n ( ). Then there is a positive continuous function φ g (l) of l ≥  depending on g such that is a solution of the Dirichlet-Schrödinger problem in H n ( ) with g.

Main lemmas
for any positive real number R. Then for any W ∈ S n ( ).

Proof of Theorem
For any fixed V = (τ , ) ∈ H n ( ), we can choose a number R satisfying R > max{, r}. Then we see from Lemma  that To see that PI a (φ g , g)(V ) is a harmonic function in H n ( ), we remark that PI a (φ g , g)(V ) satisfies the locally mean-valued property by Fubini's theorem.
Finally we shall show that lim V ∈H n ( ),V →W PI a (φ g , g)(V ) = g W for any W = (ι , ϒ ) ∈ ∂H n ( ). Setting in Lemma , which is locally integrable on S n ( ) for any fixed V ∈ H n ( ). Then we apply Lemma  to g(V ) and -g(V ). For any >  and a positive number δ, by (.) we can choose a number R (> max{, (ι + δ)}) such that (.) holds, where V ∈ H n ( ) ∩ B(W , δ).
Since where W ∈ S n ( ) and W ∈ S n ( ). Then (.) holds. Thus we complete the proof of the theorem. We wish to express our genuine thanks to the anonymous referees for careful reading and excellent comments on this manuscript.