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Infinitely many sign-changing solutions for p-Laplacian equation involving the critical Sobolev exponent

Yuanze Wu* and Yisheng Huang

Author Affiliations

Department of Mathematics, Soochow University, Suzhou, Jiangsu, 215006, P.R. China

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Boundary Value Problems 2013, 2013:149  doi:10.1186/1687-2770-2013-149

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/149


Received:14 December 2012
Accepted:28 May 2013
Published:19 June 2013

© 2013 Wu and Huang; 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 study the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M2">View MathML</a> is a smooth bounded domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M4">View MathML</a> is the p-Laplacian, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M5">View MathML</a> is the critical Sobolev exponent and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a> is a parameter. By establishing a new deformation lemma, we show that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M7">View MathML</a>, then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a>, this problem has infinitely many sign-changing solutions, which extends the results obtained in (Cao et al. in J. Funct. Anal. 262: 2861-2902, 2012; Schechter and Zou in Arch. Ration. Mech. Anal. 197: 337-356, 2010).

1 Introduction

In this paper, we consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M9">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M11">View MathML</a>) is a smooth bounded domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M4">View MathML</a> is the p-Laplacian, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M5">View MathML</a> is the critical Sobolev exponent and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a> is a parameter.

The first existence result of Problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16">View MathML</a> was obtained by Brezis and Nirenberg in the celebrated paper [1]. In that paper, the authors proved that Problem (1.1) had a positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M18">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M20">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M21">View MathML</a> is the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M22">View MathML</a>. After that, many existence results have appeared for (1.1); one can see, for example, [2-7] and the references therein for case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16">View MathML</a> and [8-11] and the references therein for case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M3">View MathML</a>. In particular, in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16">View MathML</a>, the authors in [2] proved that the number of solutions of Problem (1.1) is bounded below by the number of eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M22">View MathML</a> lying in the open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M27">View MathML</a>, where S is the best Sobolev constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M28">View MathML</a> is the Lebesgue measure of Ω. In [5], the existence of infinitely many sign-changing solutions of (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16">View MathML</a> has been obtained when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a> and Ω is a ball, while it has been shown in [6] that (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16">View MathML</a> has infinitely many sign-changing radial solutions when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a> and Ω also is a ball. We remark that the methods used in [5,6] are strongly dependent on the symmetry of the ball Ω. For a general bounded smooth domain Ω, by the method of invariant sets of the descending flow, the authors in [7] have shown that (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M16">View MathML</a> has infinitely many sign-changing solutions when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M33">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a>, which extends the main result in [4].

The main purpose of this paper is to try to obtain the existence of infinitely many sign-changing solutions of Problem (1.1) for general <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M38">View MathML</a>. In a very recent work [9], the authors have proved that (1.1) has infinitely many solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M7">View MathML</a>. However, by using the Picone identity (cf.[12,13]), we see that every nonzero solution of Problem (1.1) is sign-changing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M41">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M42">View MathML</a> is the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M43">View MathML</a> (see Lemma 2.1 for more details). Hence, to achieve our purpose, we mainly consider the situation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M44">View MathML</a>.

Our main result in this paper is the following.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M7">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a>. Then Problem (1.1) has infinitely many sign-changing solutions.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M47">View MathML</a> is the critical Sobolev exponent, in order to overcome the lack of compactness of the embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a> in the Lebesgue space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M49">View MathML</a>, we follow the ideas of [4,7,9] to consider the following auxiliary problems:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M52">View MathML</a> is increasing to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M47">View MathML</a>. It has been shown by [[14], Theorem 1.2] that for every n, Problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54">View MathML</a>) has infinitely many sign-changing solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M55">View MathML</a>. Hence, to prove Theorem 1.1, we will show that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M57">View MathML</a> converges to some sign-changing solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58">View MathML</a> of (1.1) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59">View MathML</a>, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M60">View MathML</a> are different. The convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M57">View MathML</a> can be done with the help of [[9], Theorem 1.2], which we show in Lemma 2.3. To distinguish <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M60">View MathML</a>, we shall establish a new deformation lemma on special sets in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a>; see Lemma 2.5 for details.

Throughout this paper, we will always respectively denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M65">View MathML</a> by the usual norm in spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M67">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M68">View MathML</a>). Let C be indiscriminately used to denote various positive constants.

2 Proof of Theorem 1.1

We first consider the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M41">View MathML</a>. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M42">View MathML</a>, the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M71">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M72">View MathML</a>, given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M73">View MathML</a>, is simple and there exists a positive eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M74">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M42">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M76">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M77">View MathML</a> (cf.[15]). Moreover, by [[16], Proposition 2.1], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M78">View MathML</a>. On the other hand, we have the following proposition which is the so-called Picone identity.

Proposition 2.1 [[13], Lemma A.6]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M79">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M82">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M83">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M84">View MathML</a>

and the equality holds if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M85">View MathML</a>for some constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M86">View MathML</a>.

Lemma 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M87">View MathML</a>is a nonzero solution of (1.1) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M88">View MathML</a>. Thenuis sign-changing.

Proof By a contradiction, we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M80">View MathML</a>. By using a standard regularity argument and [[17], Lemmas 3.2 and 3.3], we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M90">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M91">View MathML</a>. Thus, it follows from the strong maximum principle (cf.[18]) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M92">View MathML</a>. Now, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M93">View MathML</a>, by applying the above Picone identity (i.e., Proposition 2.1) to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M95">View MathML</a>, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M96">View MathML</a>

Noting that u is a solution of (1.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M97">View MathML</a>

It follows from the Fatou lemma that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M98">View MathML</a>

which is impossible since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M41">View MathML</a>. Therefore, we have proved Lemma 2.1. □

Next, we consider the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M103">View MathML</a>.

It is clear that the corresponding functional of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M105">View MathML</a>, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M106">View MathML</a>

is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M107">View MathML</a> Fréchet differentiable. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M108">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M109">View MathML</a> is a linearly independent sequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a>. It is easy to show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M111">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M112">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M113">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M114">View MathML</a> (cf. [[14], Lemma 3.9]). We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M115">View MathML</a>

Recall that the genus of a symmetric set A of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M117">View MathML</a>

Here, we say that A is symmetric if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M118">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M119">View MathML</a>.

By [[14], Theorem 1.2], we know that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M121">View MathML</a> has infinitely many critical points, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M55">View MathML</a>, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M123">View MathML</a> for μ small enough. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M124">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M125">View MathML</a>.

Lemma 2.2For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M127">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M128">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>.

Proof Consider the following auxiliary functional:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M130">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M131">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M132">View MathML</a>, we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M133">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M135">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>. This means

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M137">View MathML</a>

(2.2)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M138">View MathML</a> is the corresponding functional of the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M139">View MathML</a>

and the nonlinearity satisfies the assumptions of [[14], Theorem 1.2]. Thus, this equation has a sequence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M140">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M141">View MathML</a>

for μ small enough. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>, the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M144">View MathML</a>, together with (2.2), imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M145">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>. On the other hand, since for every n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M55">View MathML</a> is a sequence of solutions for (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54">View MathML</a>) whose energies satisfy (2.1), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M149">View MathML</a>. We complete the proof by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M150">View MathML</a>. □

By Lemma 2.2 and [[9], Theorem 1.2], we know that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M152">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a>. The next lemma will give more information about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58">View MathML</a>.

Lemma 2.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58">View MathML</a>is a sign-changing solution of Problem (1.1) for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>.

Proof We first prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58">View MathML</a> is a solution of Problem (1.1) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M164">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M165">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M167">View MathML</a>. If we can prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M168">View MathML</a>

(2.3)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M167">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58">View MathML</a> is a solution of (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M172">View MathML</a> is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54">View MathML</a>). Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153">View MathML</a> a.e. in Ω as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a>. By the Egoroff theorem, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M178">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M179">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M182">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M183">View MathML</a> is the Lebesgue measure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M179">View MathML</a>. This, together with the Lebesgue dominated convergence theorem, implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M185">View MathML</a>

(2.4)

On the other hand, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M167">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M187">View MathML</a>

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M93">View MathML</a>, by the above inequality and the absolute continuity of the integral, we can take δ small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M189">View MathML</a>

For this δ, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M192">View MathML</a>

for n large enough. By (2.4), for this δ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M193">View MathML</a>

for n large enough. So (2.3) holds. Moreover, by a similar proof, we can show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M194">View MathML</a>.

Next, we will show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58">View MathML</a> is sign-changing for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>. Since for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M172">View MathML</a> is a sign-changing solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54">View MathML</a>), multiplying (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M54">View MathML</a>) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M201">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M202">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M203">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M103">View MathML</a>, by the Sobolev imbedding theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M205">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M49">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M208">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M153">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59">View MathML</a>. This gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M212">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M213">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M216">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M217">View MathML</a>

Thanks to Lemma 2.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M218">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M219">View MathML</a>. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M220">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M221">View MathML</a>. Indeed, if not, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M222">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M223">View MathML</a> without loss of generality. On the one hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M58">View MathML</a> is a solution of (1.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M225">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M226">View MathML</a>. On the other hand, by [[17], Lemma 3.7], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M227">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M228">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M229">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M230">View MathML</a>. Note that by a similar proof of [[14], Lemma 3.3], we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M231">View MathML</a> for μ small enough. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M232">View MathML</a> for k large enough. This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M233">View MathML</a>

for k large enough, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M234">View MathML</a>. For the sake of convenience, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M237">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M240">View MathML</a>. Note that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M242">View MathML</a> is compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a> (cf. [[9], Theorem 1.2]). It follows from [[19], Proposition 7.5] that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M93">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M245">View MathML</a>

(2.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M246">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M247">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M248">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249">View MathML</a> small enough, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M250">View MathML</a>, then we have the following.

Lemma 2.4Assume that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M252">View MathML</a>fornlarge. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M253">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M254">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M255">View MathML</a>and largen.

Proof Assume a contradiction. Then, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M257">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M258">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M259">View MathML</a> satisfies the (PS) condition for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>. Hence there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M261">View MathML</a> such that, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M262">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M223">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M266">View MathML</a>. This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M267">View MathML</a>

Thus, by [[9], Theorem 1.2], up to a subsequence, we see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M268">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M269">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M48">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M59">View MathML</a>. Moreover, by using the arguments in the proof of Lemma 2.3, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M272">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M273">View MathML</a>. On the other hand, for large n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M274">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M275">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M276">View MathML</a>. This contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M277">View MathML</a>. □

Lemma 2.5Assume that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M253">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M279">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M255">View MathML</a>and largen. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249">View MathML</a>and an odd continuous map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M282">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M283">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M284">View MathML</a>for largen.

Proof We first assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M228">View MathML</a>. It is clear that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M286">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M287">View MathML</a>

(2.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M288">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M289">View MathML</a> be the local Lipschitz continuous operator obtained in [[14], Lemma 2.1] and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290">View MathML</a> be the solution of the following O.D.E.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M291">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M292">View MathML</a> to be the maximal interval of existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290">View MathML</a>.

Claim 1: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290">View MathML</a> cannot enter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M295">View MathML</a> before it enters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M296">View MathML</a> for small δ, large n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M297">View MathML</a>.

Indeed, if the claim fails, then for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290">View MathML</a> will enter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M295">View MathML</a> before it enters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M296">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M302">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M303">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M304">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M305">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M306">View MathML</a>

By [[14], Lemma 2.1], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M307">View MathML</a>. On the other hand, by the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M308">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M309">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M310">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M311">View MathML</a>. Thanks to [[17], Lemma 3.8], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M312">View MathML</a> for large n. This, together with (2.6) and [[14], Lemma 2.1], implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M313">View MathML</a>

A contradiction with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M314">View MathML</a>.

Claim 2: There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M315">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M316">View MathML</a> for large n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M317">View MathML</a>.

If the claim is not true, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M318">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M319">View MathML</a>. We first consider the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M320">View MathML</a>. In fact, by Claim 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M321">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M322">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M319">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M324">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M325">View MathML</a> and large n, we must have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M326">View MathML</a>

(2.7)

On the other hand, by [[14], Lemma 2.1] and [[17], Lemma 5.2], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M327">View MathML</a>

This means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M328">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M329">View MathML</a>, which contradicts with (2.7). It follows that there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M330">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M316">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M332">View MathML</a>, large n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M320">View MathML</a>. Next, we consider the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M334">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M335">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M325">View MathML</a> and large n, it follows from [[14], Lemma 2.1] and [[17], Lemma 5.2] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M337">View MathML</a>

Thus, there also exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M338">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M339">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M340">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M334">View MathML</a>. Moreover, we must have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M342">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M343">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M344">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M345">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M346">View MathML</a>

Then, by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M348">View MathML</a> is continuous. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M290">View MathML</a> is odd and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M350">View MathML</a> is even, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M348">View MathML</a> is odd and it is the desired map. The situation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M230">View MathML</a> can be proved in a similar way. Therefore, we complete the proof of this lemma. □

Proof of Theorem 1.1 We first consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M41">View MathML</a>. Thanks to Lemma 2.1 and [[9], Theorem 1.1], (1.1) has infinitely many sign-changing solutions. Next, we consider the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M354">View MathML</a>. Since for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M356">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M358">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M126">View MathML</a>. It follows that two cases may occur:

Case 1: There are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M360">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M361">View MathML</a> .

In this case, Problem (1.1) has infinitely many sign-changing solutions.

Case 2: There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M362">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M363">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M364">View MathML</a>.

In this case, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M365">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M249">View MathML</a> small enough, then Problem (1.1) also has infinitely many sign-changing solutions. Otherwise, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M367">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M368">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M369">View MathML</a>. Thanks to Lemmas 2.4 and 2.5, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M282">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M371">View MathML</a> for small δ and large n. Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M372">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M364">View MathML</a>, the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M374">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M375">View MathML</a> give that there exists a large n such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M376">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M377">View MathML</a> for small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M378">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M379">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M380">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M381">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M382">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M383">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M384">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M385">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M386">View MathML</a>. By the choice of δ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M387">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M388">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M389">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M390">View MathML</a>

A contradiction. By the properties of gen, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M391">View MathML</a>

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M392">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M372">View MathML</a> is arbitrary, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M394">View MathML</a>, which contradicts with (2.5). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors typed, read and approved the final manuscript.

Acknowledgements

The work was supported by the Natural Science Foundation of China (11071180, 11171247) and College Postgraduate Research and Innovation Project of Jiangsu Province (CXZZ110082).

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