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Existence result for semilinear elliptic systems involving critical exponents

S Khademloo*, M Farzinejad and O Khazaee kohpar

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Department of Basic Sciences, Babol University of Technology, Babol, 47148-71167, Iran

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Citation and License

Boundary Value Problems 2012, 2012:119  doi:10.1186/1687-2770-2012-119

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/119


Received:14 July 2012
Accepted:4 October 2012
Published:24 October 2012

© 2012 Khademloo et al.; 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 paper we deal with the existence of a positive solution for a class of semilinear systems of multi-singular elliptic equations which involve Sobolev critical exponents. In fact, by the analytic techniques and variational methods, we prove that there exists at least one positive solution for the system.

MSC: 35J60, 35B33.

Keywords:
semilinear elliptic system; nontrivial solution; critical exponent; variational method

1 Introduction

We consider the following elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M3">View MathML</a>) is a smooth bounded domain such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M6">View MathML</a>, are different points, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M13">View MathML</a>.

We work in the product space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M14">View MathML</a>, where the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M15">View MathML</a> is the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M16">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M17">View MathML</a>.

In resent years many publications [1-3] concerning semilinear elliptic equations involving singular points and the critical Sobolev exponent have appeared. Particularly in the last decade or so, many authors used the variational method and analytic techniques to study the existence of positive solutions of systems of the form of (1.1) or its variations; see, for example, [4-8].

Before stating the main result, we clarify some terminology. Since our method is variational in nature, we need to define the energy functional of (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M14">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M19">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M20">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M21">View MathML</a>. A pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M22">View MathML</a> is said to be a solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M23">View MathML</a>, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M24">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M25">View MathML</a>

Standard elliptic arguments show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M26">View MathML</a>

The following assumptions are needed:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M34">View MathML</a>,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M35">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M36">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M37">View MathML</a> is the first eigenvalue of L, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M39">View MathML</a> are the eigenvalues of the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M40">View MathML</a>.

The quadratic from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M41">View MathML</a> is positively defined and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M42">View MathML</a>

(1.2)

Our main results are as follows.

Theorem 1.1Suppose (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>) holds. Then for any solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M44">View MathML</a>of problem (1.1), there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M45">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M46">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M47">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M48">View MathML</a>.

Theorem 1.2Suppose (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>) holds. Then for any positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M44">View MathML</a>of problem (1.1), there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M51">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M52">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M53">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M54">View MathML</a>.

Theorem 1.3Suppose (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M35">View MathML</a>) hold. Then the problem (1.1) has a positive solution.

2 Preliminaries

On <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M57">View MathML</a>, we use the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M58">View MathML</a>

Using the Young inequality, the following best constant is well defined:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M59">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M60">View MathML</a> is the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M61">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M62">View MathML</a>.

We infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M63">View MathML</a> is attained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M64">View MathML</a> by the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M65">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M66">View MathML</a>

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M13">View MathML</a>, by the Young and Hardy-Sobolev inequalities, the following constant is well defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M71">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M72">View MathML</a>

(2.2)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M73">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M76">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M77">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M79">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M80">View MathML</a> small. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M81">View MathML</a>, by [9] we have the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M82">View MathML</a>

and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M84">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M86">View MathML</a> is the volume of the unit ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M64">View MathML</a>.

3 Asymptotic behavior of solutions

Proof of Theorem 1.1 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M88">View MathML</a> is a nontrivial solution to problem (1.1). For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M89">View MathML</a> define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M90">View MathML</a>

It is not difficult to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M91">View MathML</a> and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M92">View MathML</a>

(3.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M93">View MathML</a> small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M96">View MathML</a>. Also, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M97">View MathML</a> be a cut-off function. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M98">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M99">View MathML</a>. Multiplying the first equation of (3.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M100">View MathML</a> and the second one by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M101">View MathML</a> respectively and integrating, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M102">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M103">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M104">View MathML</a>

By the Cauchy inequality and the Young inequality, we get

(3.2)

The same result holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M106">View MathML</a>.

By letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M108">View MathML</a>, we have

(3.3)

Using Caffarelli-Kohn-Nirenberg inequality [10], we infer that

(3.4)

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M111">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M112">View MathML</a>. Now, from the Hölder inequality, we deduce that

(3.5)

(3.6)

In the sequel, we have

(3.7)

By the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M116">View MathML</a>, we obtain

(3.8)

So, from (3.4) to (3.8) it follows that

(3.9)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M120">View MathML</a> to be a constant near the zero. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M121">View MathML</a>, we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M122">View MathML</a> and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M123">View MathML</a>

(3.10)

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M124">View MathML</a> is sufficiently small such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M116">View MathML</a> is a cut-off function with the properties <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M128">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M129">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M131">View MathML</a>.

Then we have the following results:

(3.11)

where we used the Hölder inequality. From (3.9) in combination with (3.11), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M133">View MathML</a>

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M134">View MathML</a>.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M138">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M140">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M141">View MathML</a>. Using (3.12) recursively, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M142">View MathML</a>

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M143">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M144">View MathML</a>. Note that the infinite sums on the right-hand side converge, then we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M145">View MathML</a>, particularly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M146">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M147">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M148">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M149">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M150">View MathML</a>. The proof is complete. □

Proof of Theorem 1.2 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M88">View MathML</a> is a positive solution to problem (1.1). For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M89">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M153">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M154">View MathML</a>

(3.13)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M155">View MathML</a> and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M156">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M157">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M158">View MathML</a>

It is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M159">View MathML</a>

(3.14)

Combining (3.13) with (3.14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M160">View MathML</a>

Therefore, by the maximum principle in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M161">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M162">View MathML</a>

Thus, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M163">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M164">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M165">View MathML</a>, we conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M166">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M167">View MathML</a>.

Similar result also holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M168">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M169">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M167">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M171">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M167">View MathML</a>. This proves the theorem. □

4 Local <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M173">View MathML</a>-condition and the existence of positive solutions

We first establish a compactness result.

Lemma 4.1Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>) holds. ThenJsatisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M173">View MathML</a>-condition for all

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M176">View MathML</a>

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M177">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M179">View MathML</a>. The standard argument shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M180">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M14">View MathML</a>.

For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M182">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M183">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M184">View MathML</a> is a solution to (1.1). Then by the concentration-compactness principle [11-13] and up to a subsequence, there exist an at most countable set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M185">View MathML</a>, a set of different points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M186">View MathML</a>, nonnegative real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M192">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M193">View MathML</a>) such that the following convergence holds in the sense of measures:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M194">View MathML</a>

By the Sobolev inequalities [10], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M195">View MathML</a>

(4.1)

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M185">View MathML</a> is finite, and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M198">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M199">View MathML</a>.

In fact, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M200">View MathML</a> be small enough for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M202">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M203">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M205">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M206">View MathML</a> be a smooth cut-off function centered at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M207">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M209">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M211">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M213">View MathML</a>. Then

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M215">View MathML</a>

By the Sobolev inequality, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M216">View MathML</a>; and then we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M198">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M199">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M185">View MathML</a> is finite.

Now, we consider the possibility of concentration at points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M220">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M193">View MathML</a>), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M200">View MathML</a> small enough that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M223">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M225">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M228">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M229">View MathML</a> be a smooth cut-off function centered at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M220">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M232">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M233">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M234">View MathML</a>. Then

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M236">View MathML</a>

(4.2)

From (4.1) and (4.2) we derive that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M193">View MathML</a>, and then either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M239">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M240">View MathML</a>. On the other hand, from the above arguments, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M241">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M242">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M243">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M245">View MathML</a>, which contradicts the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M246">View MathML</a>. On the other hand, if there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M243">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M248">View MathML</a> or there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M189">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M250">View MathML</a>, then we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M251">View MathML</a>

which contradicts our assumptions. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M252">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M253">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M57">View MathML</a>. □

First, under the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M35">View MathML</a>), we have the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M257">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M258">View MathML</a> is a minimal point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M259">View MathML</a>, and therefore a root of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M260">View MathML</a>

Lemma 4.2Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>) holds. Then we have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M262">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M263">View MathML</a>has the minimizers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M265">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M266">View MathML</a>are the extremal functions of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M267">View MathML</a>defined as in (2.2).

Proof The argument is similar to that of [6]. □

Lemma 4.3Under the assumptions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M269">View MathML</a>

Proof Suppose (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>) holds. Define the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M271">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M272">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M273">View MathML</a> as t is close to 0. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M274">View MathML</a> is attained at some finite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M275">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M276">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M277">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M278">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M279">View MathML</a> are the positive constants independent of ε. By using (1.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M280">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M281">View MathML</a>

(4.3)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M282">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M283">View MathML</a>.

From (4.3), Lemma 4.2 and Lemma 4.3, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M284">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M285">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M287">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M288">View MathML</a>

(4.4)

 □

Proof of Theorem 1.3 Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M289">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M290">View MathML</a>

Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M27">View MathML</a>) holds. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M292">View MathML</a>, from the Young and Hardy-Sobolev inequalities, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M293">View MathML</a>

and there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M80">View MathML</a> small such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M295">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M296">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M297">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M298">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M299">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M300">View MathML</a>. By the mountain-pass theorem [14], there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M301">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M303">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M253">View MathML</a>.

From Lemma 4.2 it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M305">View MathML</a>

By Lemma 4.1 there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M180">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M180">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M308">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M57">View MathML</a>. Thus, we get a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M184">View MathML</a> of J satisfying (1.1), and c is a critical value. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M311">View MathML</a>.

Replacing respectively u, ν with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M312">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M313">View MathML</a> in terms of the right-hand side of (1.1) and repeating the above process, we can get a nonnegative nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M184">View MathML</a> of (1.1). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M315">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M316">View MathML</a> by (1.1) and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M317">View MathML</a>. Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M316">View MathML</a>, we also have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M315">View MathML</a>. There, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M320">View MathML</a>. From the maximum principle, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/119/mathml/M321">View MathML</a> in Ω. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, SK, MF and OKK contributed to each part of this work equally and read and approved the final version of the manuscript.

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