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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Positivity of the infimum eigenvalue for equations of p ( x ) -Laplace type in R N

In Hyoun Kim1 and Yun-Ho Kim2*

Author Affiliations

1 Department of Mathematics, Incheon National University, Incheon, 406-772, Republic of Korea

2 Department of Mathematics Education, Sangmyung University, Seoul, 110-743, Republic of Korea

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Boundary Value Problems 2013, 2013:214  doi:10.1186/1687-2770-2013-214


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/214


Received:29 June 2013
Accepted:21 August 2013
Published:2 October 2013

© 2013 Kim and Kim; 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

We study the following elliptic equations with variable exponents

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M3">View MathML</a>

Under suitable conditions on ϕ and f, we show the existence of positivity of the infimum of all eigenvalues for the problem above, and then give an example to demonstrate our main result.

MSC: 35D30, 35J60, 35J92, 35P30, 47J10.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian; variable exponent Lebesgue-Sobolev spaces; weak solution; eigenvalue

1 Introduction

The variable exponent problems appear in a lot of applications, for example, elastic mechanics, electro-rheological fluid dynamics and image processing, etc. The study of variable mathematical problems involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-growth conditions has attracted interest and attention in recent years. We refer the readers to [1-4] and references therein.

In this paper, we are concerned with the eigenvalue problem of a class of equations of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M7">View MathML</a>

(E)

where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M8">View MathML</a> is of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M9">View MathML</a> with continuous nonconstant function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M11">View MathML</a> satisfies a Carathéodory condition. Recently, the authors in [5] obtained the positivity of the infimum of all eigenvalues for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian type subject to the Dirichlet boundary condition. As far as the authors know, there are no results concerned with the eigenvalue problem for a more general <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian type problem in the whole space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M14">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M15">View MathML</a>, the operator involved in (E) is called the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M17">View MathML</a>. The studies for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian problems have been extensively performed by many researchers in various ways; see [5-11]. In particular, by using the Ljusternik-Schnirelmann critical point theory, Fan et al.[8] established the existence of the sequence of eigenvalues of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian Dirichlet problem; see [12] for Neumann problems. Mihăilescu and Rădulescu in [13] obtained the existence of a continuous family of eigenvalues in a neighborhood of the origin under suitable conditions.

The <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian is a natural generalization of the p-Laplacian, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M21">View MathML</a> is a constant. There are a bunch of papers, for instance, [14-18] and references therein. But the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplace operator possesses more complicated nonlinearities than the p-Laplace operator, for example, it is nonhomogeneous, so a more complicated analysis has to be carefully carried out. Some properties of the p-Laplacian eigenvalue problems may not hold for a general <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian. For example, under some conditions, the infimum of all eigenvalues for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian might be zero; see [8]. The purpose of this paper is to give suitable conditions on ϕ and f to satisfy the positivity of the infimum of all eigenvalues for (E) still. This result generalizes Benouhiba’s recent result in [6] in some sense.

This paper is organized as follows. In Section 2, we state some basic results for the variable exponent Lebesgue-Sobolev spaces, which are given in [19,20]. In Section 3, we give sufficient conditions on ϕ and f to obtain the positivity of the infimum eigenvalue for the problem (E) above. Also, we present an example to illustrate our main result.

2 Preliminaries

In this section, we state some elementary properties for the variable exponent Lebesgue-Sobolev spaces, which will be used in the next section. The basic properties of the variable exponent Lebesgue-Sobolev spaces can be found from [19,20].

To make a self-contained paper, we first recall some definitions and basic properties of the variable exponent Lebesgue spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M25">View MathML</a> and the variable exponent Lebesgue-Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M26">View MathML</a>.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M27">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M28">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M29">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M30">View MathML</a>, we introduce the variable exponent Lebesgue space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M31">View MathML</a>

endowed with the Luxemburg norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M32">View MathML</a>

The dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M25">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M34">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M35">View MathML</a>. The variable exponent Lebesgue spaces are a special case of Orlicz-Musielak spaces treated by Musielak in [21].

The variable exponent Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M26">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M37">View MathML</a>

where the norm is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M38">View MathML</a>

(2.1)

Definition 2.1 The exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M39">View MathML</a> is said to be log-Hölder continuous if there is a constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M40">View MathML</a>

(2.2)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M41">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M42">View MathML</a>.

Smooth functions are not dense in the variable exponent Sobolev spaces, without additional assumptions on the exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>. Zhikov [22] gave some examples of Lavrentiev’s phenomenon for the problems with variable exponents. These examples show that smooth functions are not dense in variable exponent Sobolev spaces. However, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a> satisfies the log-Hölder continuity condition, smooth functions are dense in variable exponent Sobolev spaces, and there is no confusion in defining the Sobolev space with zero boundary values, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M45">View MathML</a>, as the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M46">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M47">View MathML</a> (see [23,24]).

Lemma 2.2[19,20]

The space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M25">View MathML</a>is a separable, uniformly convex Banach space, and its conjugate space is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M34">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M50">View MathML</a>. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M51">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M52">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M53">View MathML</a>

Lemma 2.3[19]

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M54">View MathML</a>

Then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M55">View MathML</a> (=1; <1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M56">View MathML</a> (=1; <1), respectively;

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M56">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M58">View MathML</a>;

(3) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M59">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M60">View MathML</a>.

Lemma 2.4[11]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M61">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M62">View MathML</a>for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M64">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M65">View MathML</a>, then

(1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M66">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M67">View MathML</a>;

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M68">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M69">View MathML</a>.

Lemma 2.5[23]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M70">View MathML</a>be an open, bounded set with Lipschitz boundary, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M71">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M72">View MathML</a>satisfy the log-Hölder continuity condition (2.2). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M73">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M74">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M75">View MathML</a>

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M76">View MathML</a>

and the imbedding is compact if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M77">View MathML</a>.

Lemma 2.6[25]

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M78">View MathML</a>is a Lipschitz function with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M72">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M61">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M81">View MathML</a>for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a>. Then there is a continuous embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M83">View MathML</a>.

3 Main result

In this section, we shall give the proof of the existence of the positive eigenvalue for the problem (E), by applying the basic properties of the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M26">View MathML</a>, which were given in the previous section.

Throughout this paper, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M30">View MathML</a> satisfy the log-Hölder continuity condition (2.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M87">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M88">View MathML</a>

which is equivalent to norm (2.1).

Definition 3.1 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M89">View MathML</a> is a weak solution of the problem (E) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M90">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M91">View MathML</a>.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M92">View MathML</a>

(we allow the case that one of these sets is empty). Then it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M93">View MathML</a>. We assume that:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M95">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M96">View MathML</a>.

(HJ1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M97">View MathML</a> satisfies the following conditions: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M98">View MathML</a> is measurable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M14">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M101">View MathML</a> is locally absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M102">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a>.

(HJ2) There are a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M104">View MathML</a> and a nonnegative constant b such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M105">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M107">View MathML</a>.

(HJ3) There exists a positive constant c such that the following conditions are satisfied for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M109">View MathML</a>

(3.1)

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M110">View MathML</a>. In case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M111">View MathML</a>, assume that condition (3.1) holds for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M112">View MathML</a>, and in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M113">View MathML</a>, assume that for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M112">View MathML</a> instead

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M115">View MathML</a>

(3.2)

(HJ4) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M117">View MathML</a>, the estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M118">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M119">View MathML</a>.

Let us put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M120">View MathML</a>

and define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M121">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M122">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M123">View MathML</a>[5], and its Gateaux derivative is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M124">View MathML</a>

(3.3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M125">View MathML</a> be a real-valued function. We assume that the function f satisfies the Carathéodory condition in the sense that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M126">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M128">View MathML</a> is continuous for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M130">View MathML</a>

where q is given in (H1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M131">View MathML</a>. We assume that

(F1) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M133">View MathML</a>, and there is a nonnegative measurable function m with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M134">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M135">View MathML</a>

Denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M136">View MathML</a>, it follows from (F1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M137">View MathML</a>

Define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M138">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M139">View MathML</a>

Then it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M140">View MathML</a>, and its Gateaux derivative is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M141">View MathML</a>

(3.4)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M142">View MathML</a>. Let us consider the following quantity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M143">View MathML</a>

(3.5)

For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M145">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M146">View MathML</a> satisfies a suitable condition, Benouhiba [6] proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M147">View MathML</a>. In this section, we shall generalize the conditions on f and ϕ to satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M148">View MathML</a> still.

The following lemma plays a key role in obtaining the main result in this section.

Lemma 3.2Assume that assumptions (HJ3)-(HJ4), (H1), and (F1) hold and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M149">View MathML</a>

then the functionals Φ and Ψ satisfy the following relations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M150">View MathML</a>

(3.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M151">View MathML</a>

(3.7)

Proof Applying Lemmas 2.2, 2.4 and 2.6, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M152">View MathML</a>

(3.8)

for some positive constant C. Let u in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M153">View MathML</a>. Then it follows from (HJ3), (HJ4), (3.8) and Lemma 2.3(3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M154">View MathML</a>

(3.9)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M155">View MathML</a>, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M156">View MathML</a>

Next, we show that relation (3.7) holds. From (H2), there exists a positive constant δ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M157">View MathML</a>, and thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M158">View MathML</a>

(3.10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M159">View MathML</a> be a measurable function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M160">View MathML</a>

(3.11)

holds for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M162">View MathML</a>

(3.12)

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M164">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M89">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M166">View MathML</a>. Then it follows from (F1′) and Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M167">View MathML</a>

Therefore, without loss of generality, we may suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M168">View MathML</a>. From the inequality above, by using Lemma 2.3, Lemma 2.2 and Lemma 2.4 in order, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M169">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M170">View MathML</a>

By Young’s inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M171">View MathML</a>

Using (3.11), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M172">View MathML</a>

holds for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a>. Hence it follows from Lemma 2.6 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M174">View MathML</a>

(3.13)

for some positive constant C. Therefore, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M175">View MathML</a>

From (3.10), with the inequality above, we conclude that relation (3.7) holds. □

Lemma 3.3Assume that (HJ1)-(HJ3) and (H1) hold. Then Φ is weakly lower semi-continuous, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M176">View MathML</a>inXimplies that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M177">View MathML</a>.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M176">View MathML</a> in X as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M179">View MathML</a>. Since (HJ3) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M180">View MathML</a> is strictly monotone on X, we have that Φ is convex, and so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M181">View MathML</a>

for any n. Then we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M182">View MathML</a>

The proof is complete. □

Lemma 3.4Assume that (H1) and (F1) hold. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M183">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M89">View MathML</a>, the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M185">View MathML</a>

(3.14)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M186">View MathML</a>is either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M187">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M188">View MathML</a>.

Proof Applying Lemmas 2.2, 2.4 and 2.6, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M189">View MathML</a>

for some positive constant C. □

Lemma 3.5Assume that (H1) and (F1) hold. For almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M127">View MathML</a>, the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M192">View MathML</a>

(3.15)

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M193">View MathML</a>, estimate (3.15) is obtained from (F1′) and Young’s inequality. □

Lemma 3.6Assume that (H1) and (F1) hold. Then Ψ is weakly-strongly continuous, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M176">View MathML</a>inXimplies that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M195">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M196">View MathML</a> be a sequence in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M176">View MathML</a> in X. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M196">View MathML</a> is bounded in X. By Lemma 3.4, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M199">View MathML</a>, there is a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M200">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M201">View MathML</a>

(3.16)

holds for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M202">View MathML</a>. It follows from Lemma 3.5 that the Nemytskij operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M203">View MathML</a>

is continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M204">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M205">View MathML</a>; see Theorem 1.1 in [26]. This together with Lemma 2.5 yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M206">View MathML</a>

(3.17)

Using (3.16) and (3.17), we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M207">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M208">View MathML</a>. The proof is complete. □

We are in a position to state the main result about the existence of the positive eigenvalue for the problem (E).

Theorem 3.7Assume that (HJ1)-(HJ4), (H1), (H2), and (F1) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M209">View MathML</a>is a positive eigenvalue of the problem (E). Moreover, the problem (E) has a nontrivial weak solution for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M210">View MathML</a>.

Proof It is trivial by (3.5) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M211">View MathML</a>. Suppose to the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M212">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M196">View MathML</a> be a sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M214">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M215">View MathML</a>

As in (3.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M216">View MathML</a>

for some positive constant C. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M217">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M218">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M208">View MathML</a>. Hence it follows from Lemma 3.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M220">View MathML</a>

which contradicts with the hypothesis. Hence we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M221">View MathML</a>. The analogous argument as that in the proof of Theorem 4.5 in [5] proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M209">View MathML</a> is an eigenvalue of the problem (E); see also Theorem 3.1 in [6].

Finally, we show that the problem (E) has a nontrivial weak solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M210">View MathML</a>. Notice that u is a weak solution of (E) if and only if u is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M224">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M225">View MathML</a> is fixed. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M89">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M166">View MathML</a>. With the help of (HJ3) and (HJ4), it follows from proceeding as in the proof of relation (3.13) in Lemma 3.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M228">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M229">View MathML</a>, the inequality above implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M230">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M231">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M225">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M224">View MathML</a> is coercive. Also since the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M224">View MathML</a> is weakly lower semi-continuous by Lemmas 3.3 and 3.6, we deduce that there exists a global minimizer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M235">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M224">View MathML</a> in X. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M225">View MathML</a>, we verify by definition (3.5) that there is an element ω in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M214">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M239">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M240">View MathML</a>. So we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M241">View MathML</a>

Consequently, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M242">View MathML</a>. This completes the proof. □

Now, we consider an example to demonstrate our main result in this section.

Example 3.8 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M243">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M244">View MathML</a> satisfy the log-Hölder continuity condition (2.2). Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M245">View MathML</a>, and there is a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M246">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M247">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M63">View MathML</a>. Let us consider

In this case, put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M250">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M251">View MathML</a>. Denote the quantities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M252">View MathML</a>

If conditions (H1)-(H2) hold, then we have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M253">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M209">View MathML</a> is a positive eigenvalue of the problem (E0),

(iii) the problem (E0) has a nontrivial weak solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M210">View MathML</a>,

(iv) λ is not an eigenvalue of (E0) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M256">View MathML</a>.

Proof It is clear that conditions (HJ1)-(HJ4) and (F1) hold. From the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M209">View MathML</a>, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M259">View MathML</a>

and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M260">View MathML</a>. Also, from the same argument as that in Theorem 3.7, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M148">View MathML</a>, and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M262">View MathML</a>. Applying Theorem 3.7, the conclusions (ii) and (iii) hold. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M256">View MathML</a>. Suppose that λ is an eigenvalue of the problem (E0). Then there is an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M264">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M265">View MathML</a>

By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M257">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M267">View MathML</a>

a contradiction. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

The first author was supported by the Incheon National University Research Grant in 2012, and the second author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2012R1A1A2042187) and a 2013 Research Grant from Sangmyung University.

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