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Existence of multiple solutions for the Brezis-Nirenberg-type problem with singular coefficients

Yang Yang1*, Jihui Zhang2 and Xiusong Gu3

Author Affiliations

1 School of Science, Jiangnan University, Wuxi, 214122, China

2 School of Mathematical Science, Nanjing Normal University, Nanjing, 210097, China

3 Institute of Science, PLA University of Science and Technology, Nanjing, 211101, China

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Boundary Value Problems 2012, 2012:137  doi:10.1186/1687-2770-2012-137


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/137


Received:4 September 2012
Accepted:19 September 2012
Published:26 November 2012

© 2012 Yang 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

By energy estimates and by establishing a local (PS) condition, we obtain the multiplicity of solutions to a class of Brezis-Nirenberg-type problem with singular coefficients via minimax methods and the Krasnoselskii genus theory.

Keywords:
Brezis-Nirenberg-type problem; minimax method

1 Introduction and main results

This paper is concerned with multiple solutions for the semilinear Brezis-Nirenberg-type problem with singular coefficients

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M2">View MathML</a> is a bounded smooth domain, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M9">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M11">View MathML</a> are two real parameters.

The starting point of the variational approach to the problem is the Caffarelli-Kohn-Nirenberg inequality (see [1]): There is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M13">View MathML</a>

(2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M14">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M15">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16">View MathML</a> be the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M17">View MathML</a> with respect to the weighted norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M18">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M19">View MathML</a>

From the boundedness of Ω and the standard approximation arguments, it is easy to see that (2) holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M20">View MathML</a> in the sense:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M21">View MathML</a>

(3)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M23">View MathML</a>, that is, the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M24">View MathML</a> is continuous, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M25">View MathML</a> is the weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M26">View MathML</a> space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M27">View MathML</a>

On <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16">View MathML</a>, we can define the energy functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M29">View MathML</a>

(4)

From (4), J is well defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M31">View MathML</a>. Furthermore, the critical points of J are weak solutions of problem (1).

Breiz-Nirenberg-type problems have been generalized to many situations such as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M32">View MathML</a>

(5)

Xuan et al.[2] derived the explicit formula for the extremal functions of the best embedding constant by applying the Bliss lemma [3]. They got a nontrivial solution for problem (5) including the resonant and nonresonant cases by variational methods. He and Zou [4] studied problem (5) and obtained the multiplicity of solutions with the aid of a pseudo-index theory. In [5], problem (5) has been extended to the p-Laplace case by Xuan.

The purpose of this paper is to study the multiplicity of solutions for the Breiz-Nirenberg-type problem (1) with the aid of a minimax method. We obtain multiple nontrivial solutions of (1) by proving the local (PS) condition and energy estimates.

Our main results are the following.

Theorem 1.1Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M33">View MathML</a>, then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M35">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M36">View MathML</a>, problem (1) has a sequence of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M37">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M38">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M39">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M40">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M42">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M43">View MathML</a>, problem (1) has a sequence of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M37">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M38">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M39">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M40">View MathML</a>.

2 Preliminary results

Lemma 2.1[5]

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M2">View MathML</a>is an open bounded domain with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M49">View MathML</a>boundary and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M51">View MathML</a>. The embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M52">View MathML</a>is compact if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M54">View MathML</a>.

Lemma 2.2 (Concentration compactness principle [5])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M58">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M59">View MathML</a>be the space of bounded measures on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M60">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M61">View MathML</a>is a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M62">View MathML</a>

Then there are the following statements:

(1) There exists some at most countable setI, a family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M63">View MathML</a>of distinct points in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M60">View MathML</a>, and a family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M65">View MathML</a>of positive numbers such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M66">View MathML</a>

(6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M67">View MathML</a>is the Dirac-mass of mass 1 concentrated at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M68">View MathML</a>.

(2) The following inequality holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M69">View MathML</a>

(7)

for some family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M70">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M71">View MathML</a>

(8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M72">View MathML</a>to be the best embedding constants, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M73">View MathML</a>

In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M74">View MathML</a>.

Lemma 2.3Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75">View MathML</a>is a (PS)csequence with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M33">View MathML</a>, then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M41">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M79">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75">View MathML</a>has a convergent subsequence in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M82">View MathML</a>.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M34">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M84">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75">View MathML</a>has a convergent subsequence in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M82">View MathML</a>.

Proof (1) The boundedness of (PS)c sequence.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75">View MathML</a> is a (PS)c sequence, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M89">View MathML</a>

(9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M90">View MathML</a>

(10)

So, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M91">View MathML</a>

We have the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M33">View MathML</a>, then there exists a subsequence, we still denote it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M95">View MathML</a>

From the concentration compactness principle, there exist nonnegative measures μ, ν and a countable family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M96">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M97">View MathML</a>

(2) Up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M98">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M99">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16">View MathML</a>, we may suppose, without loss of generality, that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M102">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M103">View MathML</a>

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M104">View MathML</a> is also bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M105">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M106">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M107">View MathML</a>

(11)

taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M108">View MathML</a> in (11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M109">View MathML</a>

(12)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M110">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M111">View MathML</a> in (12), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M112">View MathML</a>, then it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M113">View MathML</a>

(13)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M108">View MathML</a> in (13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M115">View MathML</a>

(14)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M116">View MathML</a> in (12), then it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M117">View MathML</a>

(15)

Thus, it implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M118">View MathML</a>

(16)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M119">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M120">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M121">View MathML</a>.

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M122">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M123">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M124">View MathML</a>

However, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M10">View MathML</a> is given, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M84">View MathML</a> so small that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M85">View MathML</a>, the last term on the right-hand side above is greater than 0, which is a contradiction. Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M11">View MathML</a> is given, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M79">View MathML</a> so small that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M80">View MathML</a>, the last term on the right-hand side above is greater than 0. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M131">View MathML</a> for each i.

Up to now, we have shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M132">View MathML</a>

So, by the Breiz-Lieb lemma,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M133">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M134">View MathML</a>. Thus, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M75">View MathML</a> strongly converges to u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16">View MathML</a>. □

3 Existence of infinitely many solutions

In this section, we use the minimax procedure to prove the existence of infinitely many solutions. Let Σ be the class of subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M137">View MathML</a>, which are closed and symmetric with respect to the origin. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M138">View MathML</a>, we define the genus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M139">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M140">View MathML</a>

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M33">View MathML</a>, then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M142">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M143">View MathML</a>

Then, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M10">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M145">View MathML</a> so small that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M146">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M147">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M148">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M150">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M148">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M153">View MathML</a>. Similarly, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M11">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M155">View MathML</a> with the property that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M157">View MathML</a> as above exist for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M158">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M159">View MathML</a>. Following the same idea as in [6-8], we consider the truncated functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M160">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M161">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M162">View MathML</a> is a nonincreasing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M163">View MathML</a> function such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M164">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M166">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M167">View MathML</a>. The main properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168">View MathML</a> are the following.

Lemma 3.1

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M169">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168">View MathML</a>is bounded below.

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M171">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M172">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M173">View MathML</a>.

(3) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M11">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M175">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168">View MathML</a>satisfies (PS)ccondition.

(4) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M10">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M180">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168">View MathML</a>satisfies (PS)ccondition.

Proof (1) and (2) are immediate. To prove (3) and (4), observe that all (PS)c sequences for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M76">View MathML</a> must be bounded. Similar to the proof of Lemma 2.3, there exists a convergent subsequence. □

Lemma 3.2Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M186">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M187">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M188">View MathML</a>

Proof Fix m and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M189">View MathML</a> be an m-dimensional subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M16">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M192">View MathML</a>, write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M193">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M196">View MathML</a>. Thus, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M197">View MathML</a>, since all the norms are equivalent, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M198">View MathML</a>

Therefore, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M199">View MathML</a> so small that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M200">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M201">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M202">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M203">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M204">View MathML</a> and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M205">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M206">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M207">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168">View MathML</a> is bounded from below. □

Lemma 3.3Letλ, βbe as in (3) or (4) of Lemma 3.1. Then all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M209">View MathML</a>are critical values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M211">View MathML</a>.

Proof It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M213">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M214">View MathML</a>. Moreover, since all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M209">View MathML</a> are critical values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M168">View MathML</a>, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M217">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M218">View MathML</a>, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M219">View MathML</a> is compact and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M220">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M221">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M222">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M223">View MathML</a>. By the deformation lemma there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M224">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M225">View MathML</a>) and an odd homeomorphism η such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M226">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M209">View MathML</a> is increasing and converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M228">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M186">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M230">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M231">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M232">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M233">View MathML</a>. By the properties of γ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M234">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M235">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M236">View MathML</a>

a contradiction, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/137/mathml/M211">View MathML</a>. □

With Lemma 3.1 to Lemma 3.3, we have proved Theorem 1.1.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

Project is supported by National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), the China Scholarship Council, Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK2012109), the Fundamental Research Funds for the Central Universities (No. JUSRP11118, JUSRP211A22) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).

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