Open Access

Dirichlet problem for the Schrödinger operator on a cone

Boundary Value Problems20122012:59

DOI: 10.1186/1687-2770-2012-59

Received: 16 February 2012

Accepted: 2 May 2012

Published: 18 June 2012

Abstract

In this article, a solution of the Dirichlet problem for the Schrödinger operator on a cone is constructed by the generalized Poisson integral with a slowly growing continuous boundary function. A solution of the Poisson integral for any continuous boundary function is also given explicitly by the Poisson integral with the generalized Poisson kernel depending on this boundary function.

MSC:31B05, 31B10.

Keywords

Dirichlet problem stationary Schrödinger equation cone

1 Introduction and results

Let R and R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq1_HTML.gif be the set of all real numbers and the set of all positive real numbers respectively. We denote the n-dimensional Euclidean space by R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif ( n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq3_HTML.gif). A point in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif is denoted by P = ( X , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq4_HTML.gif, where X = ( x 1 , x 2 , , x n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq5_HTML.gif. The Euclidean distance between two points P and Q in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif is denoted by | P Q | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq6_HTML.gif. Also | P O | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq7_HTML.gif with the origin O of R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif is simply denoted by | P | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq8_HTML.gif. The boundary and the closure of a set S in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif are denoted by S and S ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq9_HTML.gif respectively.

We introduce a system of spherical coordinates ( r , Θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq10_HTML.gif, Θ = ( θ 1 , θ 2 , , θ n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq11_HTML.gif, in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif which are related to Cartesian coordinates ( x 1 , x 2 , , x n 1 , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq12_HTML.gif by x n = r cos θ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq13_HTML.gif.

The unit sphere and the upper half unit sphere in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif are denoted by S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq14_HTML.gif and S + n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq15_HTML.gif, respectively. For simplicity, a point ( 1 , Θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq16_HTML.gif on S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq14_HTML.gif and the set { Θ ; ( 1 , Θ ) Ω } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq17_HTML.gif for a set Ω, Ω S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq18_HTML.gif, are often identified with Θ and Ω, respectively. For two sets Ξ R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq19_HTML.gif and Ω S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq18_HTML.gif, the set { ( r , Θ ) R n ; r Ξ , ( 1 , Θ ) Ω } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq20_HTML.gif in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif is simply denoted by Ξ × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq21_HTML.gif.

For P R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq22_HTML.gif and r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq23_HTML.gif, let B ( P , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq24_HTML.gif denote an open ball with a center at P and radius r in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif. S r = B ( O , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq25_HTML.gif. By C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif, we denote the set R + × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq27_HTML.gif in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq2_HTML.gif with the domain Ω on S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq14_HTML.gif. We call it a cone. We denote the sets I × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq28_HTML.gif and I × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq29_HTML.gif with an interval on R by C n ( Ω ; I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq30_HTML.gif and S n ( Ω ; I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq31_HTML.gif. By S n ( Ω ; r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq32_HTML.gif we denote C n ( Ω ) S r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq33_HTML.gif. By S n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq34_HTML.gif we denote S n ( Ω ; ( 0 , + ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq35_HTML.gif which is C n ( Ω ) { O } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq36_HTML.gif. We denote the ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq37_HTML.gif-dimensional volume elements induced by the Euclidean metric on S r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq38_HTML.gif by d S r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq39_HTML.gif.

Let A a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq40_HTML.gif denote the class of nonnegative radial potentials a ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq41_HTML.gif, i.e., 0 a ( P ) = a ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq42_HTML.gif, P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gif, such that a L loc b ( C n ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq44_HTML.gif with some b > n / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq45_HTML.gif if n 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq46_HTML.gif and with b = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq47_HTML.gif if n = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq48_HTML.gif or n = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq49_HTML.gif.

This article is devoted to the stationary Schrödinger equation
Sch a u ( P ) = Δ u ( P ) + a ( P ) u ( P ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ1_HTML.gif
(1.1)

where P C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq50_HTML.gif, Δ is the Laplace operator and a A a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq51_HTML.gif. These solutions called a-harmonic functions or generalized harmonic functions are associated with the operator Sch a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq52_HTML.gif. Note that they are (classical) harmonic functions in the case a = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq53_HTML.gif. Under these assumptions, the operator Sch a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq54_HTML.gif can be extended in the usual way from the space C 0 ( C n ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq55_HTML.gif to an essentially self-adjoint operator on L 2 ( C n ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq56_HTML.gif (see [13]). We will denote it Sch a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq52_HTML.gif as well. This last one has a Green’s function G ( Ω , a ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq57_HTML.gif. Here G ( Ω , a ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq57_HTML.gif is positive on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif and its inner normal derivative G ( Ω , a ) ( P , Q ) / n Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq58_HTML.gif. We denote this derivative by P ( Ω , a ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq59_HTML.gif, which is called the Poisson a-kernel with respect to C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif. We remark that G ( Ω , 0 ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq60_HTML.gif and P ( Ω , 0 ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq61_HTML.gif are the Green’s function and Poisson kernel of the Laplacian in C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif respectively.

Given a domain D R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq62_HTML.gif and a continuous function u on ( D ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq63_HTML.gif, we say that h is a solution of the Dirichlet problem for the Schrödinger operator on D with u if Sch a h = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq64_HTML.gif in D and
lim P D , P Q h ( P ) = u ( Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equa_HTML.gif

for every Q ( D ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq65_HTML.gif. Note that h is a solution of the classical Dirichlet problem for the Laplacian in the case a = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq53_HTML.gif.

Let Δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq66_HTML.gif be a Laplace-Beltrami operator (the spherical part of the Laplace) on Ω S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq18_HTML.gif and λ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq67_HTML.gif ( j = 1 , 2 , 3 , , 0 < λ 1 < λ 2 λ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq68_HTML.gif) be the eigenvalues of the eigenvalue problem for Δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq66_HTML.gif on Ω (see, e.g., [4], p. 41])
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equb_HTML.gif
Corresponding eigenfunctions are denoted by φ j v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq69_HTML.gif ( 1 v v j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq70_HTML.gif), where v j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq71_HTML.gif is the multiplicity of λ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq67_HTML.gif. We set λ 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq72_HTML.gif, norm the eigenfunctions in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq73_HTML.gif and φ 1 = φ 11 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq74_HTML.gif. Then there exist two positive constants d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq75_HTML.gif and d 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq76_HTML.gif such that
d 1 δ ( P ) φ 1 ( Θ ) d 2 δ ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ2_HTML.gif
(1.2)

for P = ( 1 , Θ ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq77_HTML.gif (see Courant and Hilbert [5]), where δ ( P ) = inf Q C n ( Ω ) | P Q | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq78_HTML.gif.

In order to ensure the existences of λ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq67_HTML.gif ( j = 1 , 2 , 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq79_HTML.gif). We put a rather strong assumption on Ω: if n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq80_HTML.gif, then Ω is a C 2 , α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq81_HTML.gif-domain ( 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq82_HTML.gif) on S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq14_HTML.gif surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see [6], pp. 88-89] for the definition of C 2 , α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq81_HTML.gif-domain). Then φ j v C 2 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq83_HTML.gif ( j = 1 , 2 , 3 , , 1 v v j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq84_HTML.gif) and φ 1 / n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq85_HTML.gif on Ω (here and below, / n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq86_HTML.gif denotes differentiation along the interior normal).

Hence well-known estimates (see, e.g., [7], p. 14]) imply the following inequality:
v = 1 v j φ j v ( Θ ) φ j v ( Φ ) n Φ M ( n ) j 2 n 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ3_HTML.gif
(1.3)

where the symbol M ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq87_HTML.gif denotes a constant depending only on n.

Let V j ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq88_HTML.gif and W j ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq89_HTML.gif stand, respectively, for the increasing and nonincreasing, as r + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq90_HTML.gif, solutions of the equation
Q ( r ) n 1 r Q ( r ) + ( λ j r 2 + a ( r ) ) Q ( r ) = 0 , 0 < r < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ4_HTML.gif
(1.4)

normalized under the condition V j ( 1 ) = W j ( 1 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq91_HTML.gif.

We shall also consider the class B a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq92_HTML.gif, consisting of the potentials a A a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq93_HTML.gif such that there exists a finite limit lim r r 2 a ( r ) = k [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq94_HTML.gif; moreover, r 1 | r 2 a ( r ) k | L ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq95_HTML.gif. If a B a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq96_HTML.gif, then the solutions of Equation (1.1) are continuous (see [8]).

In the rest of the article, we assume that a B a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq97_HTML.gif and we shall suppress this assumption for simplicity. Further, we use the standard notations u + = max ( u , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq98_HTML.gif, u = min ( u , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq99_HTML.gif, [ d ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq100_HTML.gif is the integer part of d and d = [ d ] + { d } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq101_HTML.gif, where d is a positive real number.

Denote
ι j , k ± = 2 n ± ( n 2 ) 2 + 4 ( k + λ j ) 2 ( j = 0 , 1 , 2 , 3 , ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equc_HTML.gif
It is known (see [9]) that in the case under consideration the solutions to Equation (1.4) have the asymptotics
V j ( r ) d 3 r ι j , k + , W j ( r ) d 4 r ι j , k , as r , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ5_HTML.gif
(1.5)

where d 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq102_HTML.gif and d 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq103_HTML.gif are some positive constants.

If a A a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq93_HTML.gif, it is known that the following expansion for the Green function G ( Ω , a ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq57_HTML.gif (see [10], Ch. 11], [1, 11])
G ( Ω , a ) ( P , Q ) = j = 0 1 χ ( 1 ) V j ( min ( r , t ) ) W j ( max ( r , t ) ) ( v = 1 v j φ j v ( Θ ) φ j v ( Φ ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equd_HTML.gif

where P = ( r , Θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq104_HTML.gif Q = ( t , Φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq105_HTML.gif r t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq106_HTML.gif and χ ( s ) = w ( W 1 ( r ) , V 1 ( r ) ) | r = s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq107_HTML.gif, is their Wronskian. The series converges uniformly if either r s t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq108_HTML.gif or t s r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq109_HTML.gif ( 0 < s < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq110_HTML.gif).

For a nonnegative integer m and two points P = ( r , Θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq104_HTML.gif, Q = ( t , Φ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq111_HTML.gif, we put
K ( Ω , a , m ) ( P , Q ) = { 0 if 0 < t < 1 , K ˜ ( Ω , a , m ) ( P , Q ) if 1 t < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Eque_HTML.gif
where
K ˜ ( Ω , a , m ) ( P , Q ) = j = 0 m 1 χ ( 1 ) V j ( r ) W j ( t ) ( v = 1 v j φ j v ( Θ ) φ j v ( Φ ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equf_HTML.gif
We introduce another function of P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gif and Q = ( t , Φ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq111_HTML.gif
G ( Ω , a , m ) ( P , Q ) = G ( Ω , a ) ( P , Q ) K ( Ω , a , m ) ( P , Q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equg_HTML.gif
The generalized Poisson kernel P ( Ω , a , m ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq112_HTML.gif ( P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gif, Q = ( t , Φ ) S n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq113_HTML.gif) with respect to C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif is defined by
P ( Ω , a , m ) ( P , Q ) = G ( Ω , a , m ) ( P , Q ) n Q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equh_HTML.gif
In fact,
P ( Ω , a , 0 ) ( P , Q ) = P ( Ω , a ) ( P , Q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equi_HTML.gif

We remark that the kernel function P ( Ω , 0 , m ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq114_HTML.gif coincides with the one in Yoshida and Miyamoto [12] (see [10], Ch. 11]).

Put
U ( Ω , a , m ; u ) ( P ) = S n ( Ω ) P ( Ω , a , m ) ( P , Q ) u ( Q ) d σ Q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equj_HTML.gif

where u ( Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq115_HTML.gif is a continuous function on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq116_HTML.gif and d σ Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq117_HTML.gif is a surface area element on S n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq34_HTML.gif.

With regard to classical solutions of the Dirichlet problem for the Laplacian, Yoshida and Miyamoto [12], Theorem 1] proved the following result.

Theorem A If u is a continuous function on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq118_HTML.gif satisfying
S n ( Ω ) | u ( t , Φ ) | 1 + t ι m + 1 , 0 + + n 1 d σ Q < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equk_HTML.gif
then U ( Ω , 0 , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq119_HTML.gif is a classical solution of the Dirichlet problem on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif with g and satisfies
lim r , P = ( r , Θ ) C n ( Ω ) r ι m + 1 , 0 + U ( Ω , 0 , m ; u ) ( P ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equl_HTML.gif

Our first aim is to give growth properties at infinity for U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gif.

Theorem 1 Let γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq121_HTML.gif (resp. γ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq122_HTML.gif), ι [ γ ] , k + + { γ } > ι 1 , k + + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq123_HTML.gif (resp. ι [ γ ] , k + { γ } > ι 1 , k + + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq124_HTML.gif) and

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equm_HTML.gif
If u is a measurable function on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq118_HTML.gif satisfying
S n ( Ω ) | u ( t , Φ ) | 1 + t ι [ γ ] , k + + { γ } d σ Q < ( resp. S n ( Ω ) | u ( t , Φ ) | ( 1 + t ι [ γ ] , k + + { γ } ] ) d σ Q < ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ6_HTML.gif
(1.6)
then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ7_HTML.gif
(1.7)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ8_HTML.gif
(1.8)

Next, we are concerned with solutions of the Dirichlet problem for the Schrödinger operator on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif.

Theorem 2 Let γ and ι m + 1 , k + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq125_HTML.gifbe as in Theorem 1. If u is a continuous function on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq118_HTML.gifsatisfying (1.6), then U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gifis a solution of the Dirichlet problem for the Schrödinger operator on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gifwith u and (1.7) (resp. (1.8)) holds.

If we take ι [ γ ] , k + + { γ } = ι m + 1 , k + + n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq126_HTML.gif, then we immediately have the following corollary, which is just Theorem A in the case a = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq53_HTML.gif.

Corollary If u is a continuous function on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq118_HTML.gif satisfying
S n ( Ω ) | u ( t , Φ ) | 1 + t ι m + 1 , k + + n 1 d σ Q < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ9_HTML.gif
(1.9)
then U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gif is a solution of the Dirichlet problem for the Schrödinger operator on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif with u and satisfies
lim r , P = ( r , Θ ) C n ( Ω ) r ι m + 1 , k + U ( Ω , a , m ; u ) ( P ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ10_HTML.gif
(1.10)

By using Corollary, we can give a solution of the Dirichlet problem for any continuous function on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq118_HTML.gif.

Theorem 3 If u is a continuous function on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq118_HTML.gifsatisfying (1.9) and h ( r , Θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq127_HTML.gifis a solution of the Dirichlet problem for the Schrödinger operator on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gifwith u satisfying
lim r , P = ( r , Θ ) C n ( Ω ) r ι m + 1 , k + h + ( P ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ11_HTML.gif
(1.11)
then
h ( P ) = U ( Ω , a , m ; u ) ( P ) + j = 0 m ( v = 1 v j d j v φ j v ( Θ ) ) V j ( r ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equn_HTML.gif

where P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gifand d j v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq128_HTML.gifare constants.

2 Lemmas

Throughout this article, let M denote various constants independent of the variables in questions, which may be different from line to line.

Lemma 1
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ12_HTML.gif
(2.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ13_HTML.gif
(2.2)
for any P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gifand any Q = ( t , Φ ) S n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq113_HTML.gifsatisfying 0 < t r 4 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq129_HTML.gif (resp. 0 < r t 4 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq130_HTML.gif);
| P ( Ω , 0 ) ( P , Q ) | M 1 t n 1 + M r | P Q | n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ14_HTML.gif
(2.3)

for any P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gifand any Q = ( t , Φ ) S n ( Ω ; ( 4 5 r , 5 4 r ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq131_HTML.gif.

Proof (2.1) and (2.2) are obtained by Kheyfits (see [10], Ch. 11]). (2.3) follows from Azarin (see [13], Lemma 4 and Remark]). □

Lemma 2 (see [1])

For a nonnegative integer m, we have
| P ( Ω , a , m ) ( P , Q ) | M ( n , m , s ) V m + 1 ( r ) W m + 1 ( t ) t φ 1 ( Θ ) φ 1 ( Φ ) n Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ15_HTML.gif
(2.4)

for any P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gifand Q = ( t , Φ ) S n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq113_HTML.gifsatisfying r s t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq108_HTML.gif ( 0 < s < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq110_HTML.gif), where M ( n , m , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq132_HTML.gifis a constant dependent of n, m and s.

Lemma 3 (see [2], Theorem 1])

If u ( r , Θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq133_HTML.gifis a solution of Equation (1.1) on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gifsatisfying

Ω u + ( r , Θ ) d S 1 = O ( r ι m , k + ) , as r , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ16_HTML.gif
(2.5)
then
u ( r , Θ ) = j = 0 m ( v = 1 v j d j v φ j v ( Θ ) ) V j ( r ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equo_HTML.gif
Lemma 4 Obviously, the conclusion of Lemma 3 holds true if (2.5) is replaced by
lim r , ( r , Θ ) C n ( Ω ) r ι m + 1 , k + u + ( r , Θ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ17_HTML.gif
(2.6)
Proof Since
V m + 1 ( r ) r ι m + 1 , k + as r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equp_HTML.gif
from (1.5) and
ι m + 1 , k + ι m , k + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equq_HTML.gif

(2.6) gives that (2.5) holds, from which the conclusion immediately follows. □

3 Proof of Theorem 1

We only prove the case γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq121_HTML.gif, the remaining case γ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq122_HTML.gif can be proved similarly.

For any ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq134_HTML.gif, there exists R ϵ > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq135_HTML.gif such that
S n ( Ω ; ( R ϵ , ) ) | u ( Q ) | 1 + t ι [ γ ] , k + + { γ } d σ Q < ϵ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ18_HTML.gif
(3.1)
The relation G ( Ω , a ) ( P , Q ) G ( Ω , 0 ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq136_HTML.gif implies this inequality (see [14])
P ( Ω , a ) ( P , Q ) P ( Ω , 0 ) ( P , Q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ19_HTML.gif
(3.2)
For 0 < s < 4 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq137_HTML.gif and any fixed point P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gif satisfying r > 5 4 R ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq138_HTML.gif, let I 1 = S n ( Ω ; ( 0 , 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq139_HTML.gif, I 2 = S n ( Ω ; [ 1 , R ϵ ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq140_HTML.gif, I 3 = S n ( Ω ; ( R ϵ , 4 5 r ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq141_HTML.gif, I 4 = S n ( Ω ; ( 4 5 r , 5 4 r ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq142_HTML.gif, I 5 = S n ( Ω ; [ 5 4 r , r s ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq143_HTML.gif, I 6 = S n ( Ω ; [ 1 , r s ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq144_HTML.gif and I 7 = S n ( Ω ; [ r s , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq145_HTML.gif, we write
U ( Ω , a , m ; u ) ( P ) i = 1 7 U Ω , a , i ( P ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equr_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equs_HTML.gif
By ι [ γ ] , k + + { γ } > ι 1 , k + + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq123_HTML.gif, (1.6), (2.1) and (3.1), we have the following growth estimates
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ20_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ21_HTML.gif
(3.4)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ22_HTML.gif
(3.5)
We obtain by ι m + 1 , k + ι [ γ ] , k + + { γ } n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq146_HTML.gif, (2.2) and (3.1)
U Ω , a , 5 ( P ) M r ι 1 , k + S n ( Ω ; [ ( 5 / 4 ) r , ) ) t ι 1 , k 1 | u ( Q ) | d σ Q M r ι 1 , k + S n ( Ω ; [ ( 5 / 4 ) r , ) ) t ι [ γ ] , k + + { γ } + ι 1 , k 1 | u ( Q ) | t ι [ γ ] , k + + { γ } d σ Q M ϵ r ι [ γ ] , k + + { γ } n + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ23_HTML.gif
(3.6)
By (2.3) and (3.2), we consider the inequality
U Ω , a , 4 ( P ) U Ω , 0 , 4 ( P ) U Ω , 0 , 4 ( P ) + U Ω , 0 , 4 ( P ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equt_HTML.gif
where
U Ω , 0 , 4 ( P ) = M I 4 t 1 n | u ( Q ) | d σ Q , U Ω , 0 , 4 ( P ) = M r I 4 | u ( Q ) | | P Q | n d σ Q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equu_HTML.gif
We first have
U Ω , 0 , 4 ( P ) = M I 4 t ι 1 , k + + ι 1 , k 1 | u ( Q ) | d σ Q M r ι 1 , k + S n ( Ω ; ( ( 4 / 5 ) r , ) ) t ι 1 , k 1 | u ( Q ) | d σ Q M ϵ r ι [ γ ] , k + + { γ } n + 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ24_HTML.gif
(3.7)

which is similar to the estimate of U Ω , a , 5 ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq147_HTML.gif.

Next, we shall estimate U Ω , 0 , 4 ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq148_HTML.gif. Take a sufficiently small positive number d 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq149_HTML.gif such that I 4 B ( P , 1 2 r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq150_HTML.gif for any P = ( r , Θ ) Π ( d 5 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq151_HTML.gif, where
Π ( d 5 ) = { P = ( r , Θ ) C n ( Ω ) ; inf z Ω | ( 1 , Θ ) ( 1 , z ) | < d 5 , 0 < r < } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equv_HTML.gif

and divide C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif into two sets Π ( d 5 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq152_HTML.gif and C n ( Ω ) Π ( d 5 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq153_HTML.gif.

If P = ( r , Θ ) C n ( Ω ) Π ( d 5 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq154_HTML.gif, then there exists a positive d 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq155_HTML.gif such that | P Q | d 5 r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq156_HTML.gif for any Q S n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq157_HTML.gif, and hence
U Ω , 0 , 4 ( P ) M I 4 t 1 n | u ( Q ) | d σ Q M ϵ r ι [ γ ] , k + + { γ } n + 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ25_HTML.gif
(3.8)

which is similar to the estimate of U Ω , 0 , 4 ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq158_HTML.gif.

We shall consider the case P = ( r , Θ ) Π ( d 5 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq151_HTML.gif. Now put
H i ( P ) = { Q I 4 ; 2 i 1 δ ( P ) | P Q | < 2 i δ ( P ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equw_HTML.gif
Since S n ( Ω ) { Q R n : | P Q | < δ ( P ) } = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq159_HTML.gif, we have
U Ω , 0 , 4 ( P ) = M i = 1 i ( P ) H i ( P ) r | u ( Q ) | | P Q | n d σ Q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equx_HTML.gif

where i ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq160_HTML.gif is a positive integer satisfying 2 i ( P ) 1 δ ( P ) r 2 < 2 i ( P ) δ ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq161_HTML.gif.

Since we see from (1.2)
r φ 1 ( Θ ) M δ ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equy_HTML.gif
for P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gif. Similar to the estimate of U Ω , 0 , 4 ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq158_HTML.gif, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equz_HTML.gif

for i = 0 , 1 , 2 , , i ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq162_HTML.gif.

So
U Ω , 0 , 4 ( P ) M ϵ r ι [ γ ] , k + + { γ } n + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ26_HTML.gif
(3.9)
We only consider U Ω , a , 6 ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq163_HTML.gif in the case m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq164_HTML.gif, since U Ω , a , 6 ( P ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq165_HTML.gif for m = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq166_HTML.gif. By the definition of K ˜ ( Ω , a , m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq167_HTML.gif, (1.3) and Lemma 2, we see
U Ω , a , 6 ( P ) M χ ( 1 ) j = 0 m j 2 n 1 q j ( r ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equaa_HTML.gif
where
q j ( r ) = V j ( r ) I 6 W j ( t ) | u ( Q ) | t d σ Q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equab_HTML.gif
To estimate q j ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq168_HTML.gif, we write
q j ( r ) q j ( r ) + q j ( r ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equac_HTML.gif
where
q j ( r ) = V j ( r ) I 2 W j ( t ) | u ( Q ) | t d σ Q , q j ( r ) = V j ( r ) S n ( Ω ; ( R ϵ , r / s ) ) W j ( t ) | u ( Q ) | t d σ Q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equad_HTML.gif
Notice that
V j ( r ) V m + 1 ( t ) V j ( t ) t M V m + 1 ( r ) r M r ι m + 1 , k + 1 ( t 1 , R ϵ < r s ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equae_HTML.gif
Thus, by ι m + 1 , k + < ι [ γ ] , k + + { γ } n + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq169_HTML.gif, (1.5) and (1.6) we conclude
q j ( r ) = V j ( r ) I 2 | u ( Q ) | V j ( t ) t n 1 d σ Q M V j ( r ) I 2 V m + 1 ( t ) t ι m + 1 , k + | u ( Q ) | V j ( t ) t n 1 d σ Q M r ι m + 1 , k + 1 R ϵ ι [ γ ] , k + + { γ } ι m + 1 , k + n + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equaf_HTML.gif
Analogous to the estimate of q j ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq170_HTML.gif, we have
q j ( r ) M ϵ r ι [ γ ] , k + + { γ } n + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equag_HTML.gif
Thus we can conclude that
q j ( r ) M ϵ r ι [ γ ] , k + + { γ } n + 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equah_HTML.gif
which yields
U Ω , a , 6 ( P ) M ϵ r ι [ γ ] , k + + { γ } n + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ27_HTML.gif
(3.10)
By ι m + 1 , k + ι [ γ ] , k + + { γ } n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq146_HTML.gif, (1.5), (2.4) and (3.1) we have
U Ω , 0 , 7 ( P ) M V m + 1 ( r ) I 7 | u ( Q ) | V m + 1 ( t ) t n 1 d σ Q M ϵ r ι [ γ ] , k + + { γ } n + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equ28_HTML.gif
(3.11)

Combining (3.3)–(3.11), we obtain that if R ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq171_HTML.gif is sufficiently large and ϵ is sufficiently small, then U ( Ω , a , m ; u ) ( P ) = o ( r ι [ γ ] , k + + { γ } n + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq172_HTML.gif as r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq173_HTML.gif, where P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gif. Then we complete the proof of Theorem 1.

4 Proof of Theorem 2

For any fixed P = ( r , Θ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq43_HTML.gif, take a number satisfying R > max ( 1 , r s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq174_HTML.gif ( 0 < s < 4 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq137_HTML.gif). By ι m + 1 , k + ι [ γ ] , k + + { γ } n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq146_HTML.gif, (1.4), (1.6) and (2.4), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equai_HTML.gif

Thus U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gif is finite for any P C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq50_HTML.gif. Since P ( Ω , a , m ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq112_HTML.gif is a generalized harmonic function of P C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq50_HTML.gif for any fixed Q S n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq157_HTML.gif, U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gif is also a generalized harmonic function of P C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq50_HTML.gif. That is to say, U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gif is a solution of Equation (1.1) on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif.

Now we study the boundary behavior of U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gif. Let Q = ( t , Φ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq175_HTML.gif be any fixed point and l be any positive number satisfying l > max ( t + 1 , 4 5 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq176_HTML.gif.

Set χ S ( l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq177_HTML.gif is a characteristic function of S ( l ) = { Q = ( t , Φ ) C n ( Ω ) , t l } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq178_HTML.gif and write
U ( Ω , a , m ; u ) ( P ) = U ( P ) U ( P ) + U ( P ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equaj_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equak_HTML.gif

Notice that U ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq179_HTML.gif is the Poisson a-integral of u ( Q ) χ S ( ( 5 / 4 ) l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq180_HTML.gif, we have lim P Q , P C n ( Ω ) U ( P ) = u ( Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq181_HTML.gif. Since lim Θ Φ φ j v ( Θ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq182_HTML.gif ( j = 1 , 2 , 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq79_HTML.gif; 1 v v j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq70_HTML.gif) as P = ( r , Θ ) Q = ( t , Φ ) S n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq183_HTML.gif, we have lim P Q , P C n ( Ω ) U ( P ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq184_HTML.gif from the definition of the kernel function K ( Ω , a , m ) ( P , Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq185_HTML.gif. U ( P ) = O ( r ι [ γ ] , k + + { γ } n + 1 φ 1 ( Θ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq186_HTML.gif, and therefore tends to zero.

So the function U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gif can be continuously extended to C n ( Ω ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq187_HTML.gif such that
lim P Q , P C n ( Ω ) U ( Ω , a , m ; u ) ( P ) = u ( Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equal_HTML.gif

for any Q = ( t , Φ ) C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq188_HTML.gif from the arbitrariness of l. Thus we complete the proof of Theorem 2 from Theorem 1.

5 Proof of Theorem 3

From Corollary, we have the solution U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq120_HTML.gif of the Dirichlet problem on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif with u satisfying (1.9). Consider the function h ( P ) U ( Ω , a , m ; u ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq189_HTML.gif. Then it follows that this is the solution of Equation (1.1) in C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq26_HTML.gif and vanishes continuously on C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq118_HTML.gif.

Since
0 ( h U ( Ω , a , m ; u ) ) + ( P ) h + ( P ) + ( U ( Ω , a , m ; u ) ) ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equam_HTML.gif
for any P C n ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_IEq50_HTML.gif, we have
lim r , P = ( r , Θ ) C n ( Ω ) r ι m + 1 , k + ( h U ( Ω , a , m ; u ) ) + ( P ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-59/MediaObjects/13661_2012_Article_164_Equan_HTML.gif

from (1.10) and (1.11). Then the conclusions of Theorem 3 follow immediately from Lemma 4.

Declarations

Acknowledgements

The authors would like to thank anonymous reviewers for their valuable comments and suggestions about improving the quality of the manuscript. This work is supported by The National Natural Science Foundation of China under Grant 11071020 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20100003110004.

Authors’ Affiliations

(1)
Department of Mathematics and Information Science, Henan University of Economics and Law
(2)
School of Mathematical Science, Laboratory of Mathematics and Complex Systems, MOE Beijing Normal University

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© Qiao and Deng; licensee Springer 2012

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