Open Access Research

Dirichlet problem for the Schrödinger operator on a cone

Lei Qiao1* and Guan-Tie Deng2

Author Affiliations

1 Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450002, P.R. China

2 School of Mathematical Science, Laboratory of Mathematics and Complex Systems, MOE Beijing Normal University, Beijing, 100875, P.R. China

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Boundary Value Problems 2012, 2012:59  doi:10.1186/1687-2770-2012-59


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/59


Received:16 February 2012
Accepted:2 May 2012
Published:18 June 2012

© 2012 Qiao and Deng; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M1">View MathML</a> be the set of all real numbers and the set of all positive real numbers respectively. We denote the n-dimensional Euclidean space by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M3">View MathML</a>). A point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M5">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M6">View MathML</a>. The Euclidean distance between two points P and Q in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M8">View MathML</a>. Also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M9">View MathML</a> with the origin O of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> is simply denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M11">View MathML</a>. The boundary and the closure of a set S in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> are denoted by S and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M13">View MathML</a> respectively.

We introduce a system of spherical coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M15">View MathML</a>, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> which are related to Cartesian coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M17">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M18">View MathML</a>.

The unit sphere and the upper half unit sphere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M21">View MathML</a>, respectively. For simplicity, a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M22">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M20">View MathML</a> and the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M24">View MathML</a> for a set Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M25">View MathML</a>, are often identified with Θ and Ω, respectively. For two sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M25">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M28">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> is simply denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M30">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M32">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M33">View MathML</a> denote an open ball with a center at P and radius r in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M35">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>, we denote the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M37">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M2">View MathML</a> with the domain Ω on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M20">View MathML</a>. We call it a cone. We denote the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M41">View MathML</a> with an interval on R by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M43">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M44">View MathML</a> we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M45">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M46">View MathML</a> we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M47">View MathML</a> which is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M48">View MathML</a>. We denote the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M49">View MathML</a>-dimensional volume elements induced by the Euclidean metric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M50">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M51">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M52">View MathML</a> denote the class of nonnegative radial potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M53">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M56">View MathML</a> with some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M57">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M58">View MathML</a> and with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M59">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M60">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M61">View MathML</a>.

This article is devoted to the stationary Schrödinger equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M62">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63">View MathML</a>, Δ is the Laplace operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M64">View MathML</a>. These solutions called a-harmonic functions or generalized harmonic functions are associated with the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M65">View MathML</a>. Note that they are (classical) harmonic functions in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M66">View MathML</a>. Under these assumptions, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M67">View MathML</a> can be extended in the usual way from the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M68">View MathML</a> to an essentially self-adjoint operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M69">View MathML</a> (see [1-3]). We will denote it <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M65">View MathML</a> as well. This last one has a Green’s function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M71">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M71">View MathML</a> is positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a> and its inner normal derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M74">View MathML</a>. We denote this derivative by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M75">View MathML</a>, which is called the Poisson a-kernel with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>. We remark that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M77">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M78">View MathML</a> are the Green’s function and Poisson kernel of the Laplacian in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a> respectively.

Given a domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M80">View MathML</a> and a continuous function u on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M81">View MathML</a>, we say that h is a solution of the Dirichlet problem for the Schrödinger operator on D with u if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M82">View MathML</a> in D and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M83">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M84">View MathML</a>. Note that h is a solution of the classical Dirichlet problem for the Laplacian in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M66">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M86">View MathML</a> be a Laplace-Beltrami operator (the spherical part of the Laplace) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M88">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M89">View MathML</a>) be the eigenvalues of the eigenvalue problem for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M86">View MathML</a> on Ω (see, e.g., [4], p. 41])

Corresponding eigenfunctions are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M92">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M93">View MathML</a>), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M94">View MathML</a> is the multiplicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M88">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M96">View MathML</a>, norm the eigenfunctions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M97">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M98">View MathML</a>. Then there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M100">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M101">View MathML</a>

(1.2)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M102">View MathML</a> (see Courant and Hilbert [5]), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M103">View MathML</a>.

In order to ensure the existences of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M88">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M105">View MathML</a>). We put a rather strong assumption on Ω: if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M106">View MathML</a>, then Ω is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M107">View MathML</a>-domain (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M108">View MathML</a>) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M20">View MathML</a> surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see [6], pp. 88-89] for the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M107">View MathML</a>-domain). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M111">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M112">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M113">View MathML</a> on Ω (here and below, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M114">View MathML</a> denotes differentiation along the interior normal).

Hence well-known estimates (see, e.g., [7], p. 14]) imply the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M115">View MathML</a>

(1.3)

where the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M116">View MathML</a> denotes a constant depending only on n.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M117">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M118">View MathML</a> stand, respectively, for the increasing and nonincreasing, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M119">View MathML</a>, solutions of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M120">View MathML</a>

(1.4)

normalized under the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M121">View MathML</a>.

We shall also consider the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M122">View MathML</a>, consisting of the potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M123">View MathML</a> such that there exists a finite limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M124">View MathML</a>; moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M125">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M126">View MathML</a>, then the solutions of Equation (1.1) are continuous (see [8]).

In the rest of the article, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M127">View MathML</a> and we shall suppress this assumption for simplicity. Further, we use the standard notations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M130">View MathML</a> is the integer part of d and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M131">View MathML</a>, where d is a positive real number.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M132">View MathML</a>

It is known (see [9]) that in the case under consideration the solutions to Equation (1.4) have the asymptotics

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M133">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M134">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M135">View MathML</a> are some positive constants.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M123">View MathML</a>, it is known that the following expansion for the Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M71">View MathML</a> (see [10], Ch. 11], [1,11])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M138">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M139">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M140">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M142">View MathML</a>, is their Wronskian. The series converges uniformly if either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M143">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M144">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M145">View MathML</a>).

For a nonnegative integer m and two points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M147">View MathML</a>, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M148">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M149">View MathML</a>

We introduce another function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M147">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M152">View MathML</a>

The generalized Poisson kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M153">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M155">View MathML</a>) with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M157">View MathML</a>

In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M158">View MathML</a>

We remark that the kernel function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M159">View MathML</a> coincides with the one in Yoshida and Miyamoto [12] (see [10], Ch. 11]).

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M160">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M161">View MathML</a> is a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M163">View MathML</a> is a surface area element on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M46">View MathML</a>.

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

Theorem AIfuis a continuous function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M166">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M167">View MathML</a>is a classical solution of the Dirichlet problem on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>withgand satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M169">View MathML</a>

Our first aim is to give growth properties at infinity for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a>.

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M171">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M172">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M173">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M174">View MathML</a>) and

Ifuis a measurable function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M177">View MathML</a>

(1.6)
then

(1.7)

(1.8)

Next, we are concerned with solutions of the Dirichlet problem for the Schrödinger operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>.

Theorem 2Letγand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M181">View MathML</a>be as in Theorem 1. Ifuis a continuous function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165">View MathML</a>satisfying (1.6), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a>is a solution of the Dirichlet problem for the Schrödinger operator on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>withuand (1.7) (resp. (1.8)) holds.

If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M185">View MathML</a>, then we immediately have the following corollary, which is just Theorem A in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M66">View MathML</a>.

CorollaryIfuis a continuous function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M188">View MathML</a>

(1.9)
then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a>is a solution of the Dirichlet problem for the Schrödinger operator on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>withuand satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M191">View MathML</a>

(1.10)

By using Corollary, we can give a solution of the Dirichlet problem for any continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165">View MathML</a>.

Theorem 3Ifuis a continuous function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165">View MathML</a>satisfying (1.9) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M194">View MathML</a>is a solution of the Dirichlet problem for the Schrödinger operator on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>withusatisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M196">View MathML</a>

(1.11)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M197">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M199">View MathML</a>are 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

(2.1)

(2.2)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M155">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M204">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M205">View MathML</a>);

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M206">View MathML</a>

(2.3)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M208">View MathML</a>.

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 integerm, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M209">View MathML</a>

(2.4)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M155">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M143">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M145">View MathML</a>), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M214">View MathML</a>is a constant dependent ofn, mands.

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

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M215">View MathML</a>is a solution of Equation (1.1) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M217">View MathML</a>

(2.5)
then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M218">View MathML</a>

Lemma 4Obviously, the conclusion of Lemma 3 holds true if (2.5) is replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M219">View MathML</a>

(2.6)

Proof Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M220">View MathML</a>

from (1.5) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M221">View MathML</a>

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

3 Proof of Theorem 1

We only prove the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M171">View MathML</a>, the remaining case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M172">View MathML</a> can be proved similarly.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M224">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M225">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M226">View MathML</a>

(3.1)

The relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M227">View MathML</a> implies this inequality (see [14])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M228">View MathML</a>

(3.2)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M229">View MathML</a> and any fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M231">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M237">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M238">View MathML</a>, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M239">View MathML</a>

where

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M173">View MathML</a>, (1.6), (2.1) and (3.1), we have the following growth estimates

(3.3)

(3.4)

(3.5)

We obtain by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M245">View MathML</a>, (2.2) and (3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M246">View MathML</a>

(3.6)

By (2.3) and (3.2), we consider the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M247">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M248">View MathML</a>

We first have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M249">View MathML</a>

(3.7)

which is similar to the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M250">View MathML</a>.

Next, we shall estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M251">View MathML</a>. Take a sufficiently small positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M252">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M253">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M254">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M255">View MathML</a>

and divide <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a> into two sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M258">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M259">View MathML</a>, then there exists a positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M260">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M261">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M262">View MathML</a>, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M263">View MathML</a>

(3.8)

which is similar to the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M264">View MathML</a>.

We shall consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M254">View MathML</a>. Now put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M266">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M267">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M268">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M269">View MathML</a> is a positive integer satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M270">View MathML</a>.

Since we see from (1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M271">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>. Similar to the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M264">View MathML</a>, we obtain

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M275">View MathML</a>.

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M276">View MathML</a>

(3.9)

We only consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M277">View MathML</a> in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M278">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M279">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M280">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M281">View MathML</a>, (1.3) and Lemma 2, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M282">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M283">View MathML</a>

To estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M284">View MathML</a>, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M285">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M286">View MathML</a>

Notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M287">View MathML</a>

Thus, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M288">View MathML</a>, (1.5) and (1.6) we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M289">View MathML</a>

Analogous to the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M290">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M291">View MathML</a>

Thus we can conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M292">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M293">View MathML</a>

(3.10)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M245">View MathML</a>, (1.5), (2.4) and (3.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M295">View MathML</a>

(3.11)

Combining (3.3)–(3.11), we obtain that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M296">View MathML</a> is sufficiently large and ϵ is sufficiently small, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M297">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M298">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>. Then we complete the proof of Theorem 1.

4 Proof of Theorem 2

For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M55">View MathML</a>, take a number satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M301">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M229">View MathML</a>). By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M245">View MathML</a>, (1.4), (1.6) and (2.4), we have

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a> is finite for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M153">View MathML</a> is a generalized harmonic function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63">View MathML</a> for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a> is also a generalized harmonic function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63">View MathML</a>. That is to say, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a> is a solution of Equation (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a>.

Now we study the boundary behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M315">View MathML</a> be any fixed point and l be any positive number satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M316">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M317">View MathML</a> is a characteristic function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M318">View MathML</a> and write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M319">View MathML</a>

where

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M321">View MathML</a> is the Poisson a-integral of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M322">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M323">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M324">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M105">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M93">View MathML</a>) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M327">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M328">View MathML</a> from the definition of the kernel function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M329">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M330">View MathML</a>, and therefore tends to zero.

So the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a> can be continuously extended to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M332">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M333">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M334">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M170">View MathML</a> of the Dirichlet problem on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a> with u satisfying (1.9). Consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M337">View MathML</a>. Then it follows that this is the solution of Equation (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M36">View MathML</a> and vanishes continuously on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M165">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M340">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M63">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/59/mathml/M342">View MathML</a>

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

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

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.

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