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Existence of an unbounded branch of the set of solutions for equations of p(x)-Laplace type

Yun-Ho Kim

Author Affiliations

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

Boundary Value Problems 2014, 2014:27  doi:10.1186/1687-2770-2014-27

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/27


Received:2 October 2013
Accepted:9 January 2014
Published:30 January 2014

© 2014 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 are concerned with the following nonlinear problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M2">View MathML</a>

subject to Dirichlet boundary conditions, provided that μ is not an eigenvalue of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian. The purpose of this paper is to study the global behavior of the set of solutions for nonlinear equations of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian type by applying a bifurcation result for nonlinear operator equations.

MSC: 35B32, 35D30, 35J60, 35P30, 37K50, 46E35, 47J10.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian; variable exponent Lebesgue-Sobolev spaces; weak solution; eigenvalue

1 Introduction

Rabinowitz [1] showed that the bifurcation occurring in the Krasnoselskii theorem is actually a global phenomenon by using the topological approach of Krasnoselskii [2]. As regards the p-Laplacian and generalized operators, nonlinear eigenvalue and bifurcation problems have been extensively studied by many researchers in various ways of approach; see [3-9]. While most of those results considered global branches bifurcating from the principal eigenvalue of the p-Laplacian, under suitable conditions, Väth [10] introduced another new approach to establish the existence of a global branch of solutions for the p-Laplacian problems by using nonlinear spectral theory for homogeneous operators. Recently, Kim and Väth [11] proposed a new approach. They observed the asymptotic behavior of an integral operator corresponding to the nonhomogeneous principal part at infinity and established the existence of an unbounded branch of solutions for equations involving nonhomogeneous operators of p-Laplace type.

In recent years, the study of differential equations and variational problems involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-growth conditions has received considerable attention since they can model physical phenomena which arise in the study of elastic mechanics, electro-rheological fluid dynamics and image processing, etc. We refer the readers to [12-15] and the references therein.

In this paper, we are concerned with the existence of an unbounded branch of the set of solutions for nonlinear elliptic equations of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian type subject to the Dirichlet boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M8">View MathML</a>

(B)

when μ is not an eigenvalue of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M9">View MathML</a>

(E)

Here Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M10">View MathML</a> with Lipschitz boundary Ω, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M11">View MathML</a> are of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M12">View MathML</a> with a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M14">View MathML</a> satisfies a Carathéodory condition. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a> is a constant function, the existence of an unbounded branch of the set of solutions for equations of p-Laplacian type operator is obtained in [11] (for generalizations to unbounded domains with weighted functions, see also [16,17]). For the case of a variable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>, the authors in [18] obtained the global bifurcation result for a class of degenerate elliptic equations by observing some properties of the corresponding integral operators in the weighted variable exponent Lebesgue-Sobolev spaces.

In the particular case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M17">View MathML</a>, the operator involved in (B) is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian. The studies for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian problems have been extensively considered by many researchers in various ways; see [18-23]. As far as we know, there are no papers concerned with the bifurcation theory for the nonlinear elliptic equations involving variable exponents except [18]. Noting that (B) has more complicated nonlinearities (it is nonhomogeneous) than the p-Laplacian equation, we need some more careful and new estimates. In particular, the fact that the principal eigenvalue for problem (E) is isolated plays a key role in obtaining the bifurcation result from the principal eigenvalue. Unfortunately, under some conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>, the infimum of all positive eigenvalues for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian might be zero; see [21]. This means that there is no principal eigenvalue for some variable exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>. Even if there exists a principal eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M23">View MathML</a>, this may not be isolated because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M23">View MathML</a> is the infimum of all positive eigenvalues. Thus we cannot investigate the existence of global branches bifurcating from the principal eigenvalue of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian. However, based on the work of Väth [10], global behavior of solutions for nonlinear problems involving the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>-Laplacian was considered in [18].

This paper is organized as follows. In Section 2, we state some basic results for the variable exponent Lebesgue-Sobolev spaces. In Section 3, some properties of the corresponding integral operators are presented. We will prove the main result on global bifurcation for problem (B) in Section 4. Finally, we give an example to illustrate our bifurcation 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 sections. The basic properties of the variable exponent Lebesgue-Sobolev spaces can be found in [24,25].

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/2014/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M27">View MathML</a> and the variable exponent Lebesgue-Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M28">View MathML</a>.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M29">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M30">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M31">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M32">View MathML</a>, we introduce the variable exponent Lebesgue space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M33">View MathML</a>

endowed with the Luxemburg norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M34">View MathML</a>

The dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M27">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M36">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M37">View MathML</a>. The variable exponent Lebesgue spaces are a special case of Orlicz-Musielak spaces treated by Musielak in [26].

The variable exponent Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M28">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M39">View MathML</a>

where the norm is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M40">View MathML</a>

(2.1)

Definition 2.1 The exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M41">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/2014/1/27/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M42">View MathML</a>

(2.2)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M43">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M44">View MathML</a>.

Without additional assumptions on the exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>, smooth functions are not dense in the variable exponent Sobolev spaces. This was considered by Zhikov [27] in connection with Lavrentiev phenomenon. The importance of this above notion relies on the following fact: if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a> is log-Hölder continuous, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M47">View MathML</a> is dense in the variable exponent Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M28">View MathML</a> (see [28,29]).

Lemma 2.2 ([24,25])

The space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M27">View MathML</a>is a separable, uniformly convex Banach space, and its conjugate space is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M36">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M51">View MathML</a>. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M52">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M53">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M54">View MathML</a>

Lemma 2.3 ([24])

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M55">View MathML</a>

Then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M56">View MathML</a> (=1; <1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M57">View MathML</a> (=1; <1), respectively;

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M58">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M59">View MathML</a>;

(3) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M60">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M61">View MathML</a>.

Lemma 2.4 ([23])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M62">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M63">View MathML</a>for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M65">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M66">View MathML</a>, then

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M67">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M68">View MathML</a>;

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M69">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M70">View MathML</a>.

Lemma 2.5 ([20])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M71">View MathML</a>satisfy the log-Hölder continuity condition (2.2). Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M72">View MathML</a>, the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M41">View MathML</a>-Poincaré inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M74">View MathML</a>

holds, where the positive constantCdepends onpand Ω.

Lemma 2.6 ([28])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M75">View MathML</a>be an open, bounded set with Lipschitz boundary and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M32">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M77">View MathML</a>satisfy the log-Hölder continuity condition (2.2). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M62">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M79">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M80">View MathML</a>

(2.3)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M82">View MathML</a>

and the imbedding is compact if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M83">View MathML</a>.

3 Properties of the integral operators

In this section, we shall give some properties of the integral operators corresponding to problem (B) by applying the basic properties of the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M28">View MathML</a> which were given in the previous section.

Throughout this paper, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M32">View MathML</a> satisfy the log-Hölder continuity condition (2.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M87">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M88">View MathML</a>

which is equivalent to norm (2.1) due to Lemma 2.5.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M89">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/2014/1/27/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M90">View MathML</a>. We assume that

(HJ1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M91">View MathML</a> satisfies the following conditions: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M92">View MathML</a> is measurable on Ω for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M94">View MathML</a> is locally absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M95">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M96">View MathML</a>.

(HJ2) There are a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M97">View MathML</a> and a nonnegative constant b such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M98">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M100">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/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M102">View MathML</a>

(3.1)

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M103">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M104">View MathML</a>, then condition (3.1) holds for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M105">View MathML</a>, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M106">View MathML</a>, then assume for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M105">View MathML</a> instead

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M108">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M109">View MathML</a> denote the usual of X and its dual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110">View MathML</a> or the Euclidean scalar product on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M10">View MathML</a>, respectively. Under hypotheses (HJ1) and (HJ2), we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M112">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M113">View MathML</a>

(3.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M114">View MathML</a>.

The following estimate is a starting point for obtaining that the operator J is a homeomorphism. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a> is constant, this is a particular form of Corollary 3.1 in [11] which is based on Lemma 3.1 in [6]; see [[30], Lemma 1]. In fact, the special case that ϕ is independent of x is considered in [6]. The proof of the following proposition is essentially the same as that in [31]. For convenience, we give the proof.

Proposition 3.1Let (HJ1) and (HJ3) be satisfied. Then the following estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M116">View MathML</a>

(3.4)

holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M117">View MathML</a>, wherecis the positive constant from (HJ3).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M118">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M119">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M120">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M121">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M122">View MathML</a>. Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M123">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M124">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M104">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M126">View MathML</a>, it follows from (3.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M127">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M128">View MathML</a>

(3.5)

Noticing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M129">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M130">View MathML</a>, we have by (3.5) and (3.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M131">View MathML</a>

Without loss of generality, we may suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M132">View MathML</a>. Then we obtain, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M133">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M134">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M135">View MathML</a>

Now assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M106">View MathML</a>. As before, we obtain from (3.1) and (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M137">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M124">View MathML</a>. Using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M139">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M140">View MathML</a>

This completes the proof. □

From Proposition 3.1, we can obtain the following result.

Theorem 3.2Assume that (HJ1)-(HJ3) hold. Then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M141">View MathML</a>is a continuous, bounded, strictly monotone and coercive onX.

Proof In view of (HJ1) and (HJ2), the superposition operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M142">View MathML</a>

acts from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M143">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M144">View MathML</a> and is continuous; see Corollary 5.2.1 in [32]. Hence the continuity of J follows from the fact that J is the composition of the continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M145">View MathML</a>, the map Λ and the bounded linear map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M146">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M147">View MathML</a>

Hence the operator J is bounded and continuous on X.

For any u in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M148">View MathML</a>, it follows from (HJ3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M149">View MathML</a>

for some positive constant C. Thus we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M150">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M151">View MathML</a> and therefore the operator J is coercive on X.

Next we will show that the operator J is strictly monotone on X. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M152">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M153">View MathML</a>

(Of course, if the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M155">View MathML</a> are nonempty, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M156">View MathML</a> by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a>.) It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M158">View MathML</a>

To get strict monotonicity of the operator J, without loss of generality, we divide the proof into two cases.

Case 1. Let u, v be in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M159">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M160">View MathML</a>. By Proposition 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M161">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M162">View MathML</a>. Integrating the above inequality over Ω and using Lemma 2.3, we assert that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M163">View MathML</a>

(3.7)

For almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M164">View MathML</a>, by Proposition 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M165">View MathML</a>

(3.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M166">View MathML</a>. From Hölder’s inequality in Lemma 2.2, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M167">View MathML</a>

(3.9)

The first term on the right-hand side in (3.9) is calculated by Lemma 2.4 as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M168">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M169">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M170">View MathML</a>, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M171">View MathML</a>

(3.10)

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M172">View MathML</a>

then it follows from (3.9), (3.10), Lemmas 2.3 and 2.4 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M173">View MathML</a>

Hence we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M174">View MathML</a>

(3.11)

On the other hand, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M175">View MathML</a>

then the analogous argument implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M176">View MathML</a>

(3.12)

for some positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M177">View MathML</a>. From the previous inequalities (3.11) and (3.12), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M178">View MathML</a>

(3.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M179">View MathML</a> is a positive constant. Consequently, we obtain by (3.7) and (3.13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M180">View MathML</a>

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M182">View MathML</a>.

Case 2. Let u, v be in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M183">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M185">View MathML</a>. For almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M164">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M187">View MathML</a>

(3.14)

From the above relation (3.14) and Lemmas 2.2 and 2.4, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M188">View MathML</a>

(3.15)

where α is either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M189">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M190">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M191">View MathML</a>, we assert by (3.15) and Proposition 3.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M192">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M193">View MathML</a>

(3.16)

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M164">View MathML</a>. For almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M162">View MathML</a>, Proposition 3.1 yields the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M196">View MathML</a>

(3.17)

Consequently, it follows from (3.16) and (3.17) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M197">View MathML</a>

(3.18)

for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M198">View MathML</a>. This completes the proof. □

Using the previous result, we show the topological property of the operator J which will be needed in the main result of the next section.

Lemma 3.3If (HJ1), (HJ2) and (HJ3) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M199">View MathML</a>is a homeomorphism onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110">View MathML</a>.

Proof From Theorem 3.2, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M141">View MathML</a> is strictly monotone and coercive. The Browder-Minty theorem hence implies that the inverse operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M202">View MathML</a> exists and is bounded; see Theorem 26.A. in [33]. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M203">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M204">View MathML</a> be any sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110">View MathML</a> that converges to h in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M207">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M208">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M209">View MathML</a>. We obtain from (3.18) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M210">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M211">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M212">View MathML</a> is bounded in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M213">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M110">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M215">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M216">View MathML</a> converges to u in X. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M217">View MathML</a> is continuous at each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M203">View MathML</a>. This completes the proof. □

The main idea in obtaining our bifurcation result is to study the asymptotic behavior of the integral operator J and then to deduce a spectral result for operators that are not necessarily homogeneous. To do this, we consider a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M219">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M220">View MathML</a>

and an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M221">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M222">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M114">View MathML</a>.

To discuss the asymptotic behavior of J, we require the following hypothesis.

(HJ4) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M224">View MathML</a>, there is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M225">View MathML</a> such that for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M226">View MathML</a> the following holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M227">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M228">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M229">View MathML</a>.

Now we can show that the operators J and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M230">View MathML</a> are asymptotic at infinity, as in Proposition 5.1 of [11].

Proposition 3.4Assume that (HJ1), (HJ2) and (HJ4) are fulfilled. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M231">View MathML</a>

Proof Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M224">View MathML</a>, choose an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M233">View MathML</a> such that for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M226">View MathML</a> the following holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M235">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M228">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M237">View MathML</a>. We have by (HJ2) that for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M226">View MathML</a> the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M239">View MathML</a>

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M100">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M241">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M242">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M243">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M244">View MathML</a> and for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M226">View MathML</a>, the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M246">View MathML</a>

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M100">View MathML</a>. From Hölder’s inequality, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M248">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M114">View MathML</a>, and hence for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M250">View MathML</a>, we obtain by Minkowski’s inequality and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M251">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M252">View MathML</a>

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M179">View MathML</a>. From

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M255">View MathML</a>

the conclusion follows, because the right-hand side of the inequality tends to ε as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M151">View MathML</a>. This completes the proof. □

Next we deal with the properties for the superposition operator induced by the function f in (B). In particular, we give the compactness of this operator and the behavior of that at infinity, respectively. The ideas of the proof about these properties are completely the same as in [18]. We assume that the variable exponents are subject to the following restrictions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M257">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>. Assume that

(F1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M259">View MathML</a> satisfies the Carathéodory condition in the sense that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M260">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M262">View MathML</a> is continuous for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>.

(F2) For each bounded interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M264">View MathML</a>, there are a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M265">View MathML</a> and a nonnegative constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M266">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M267">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M269">View MathML</a>.

(F3) There exist a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M97">View MathML</a> and a locally bounded function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M271">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M272">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M273">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M275">View MathML</a>.

Under assumptions (F1) and (F2), we can define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M276">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M277">View MathML</a>

(3.19)

and an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M278">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M279">View MathML</a>

(3.20)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M280">View MathML</a>.

In proving the following result, a key idea is to use a continuity result on the superposition operators due to Väth [34]. For the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a> is a constant function, it has been proved in [11].

Lemma 3.5If (F1) and (F2) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M282">View MathML</a>is continuous and compact. Moreover, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M278">View MathML</a>is continuous and compact.

Proof A linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M284">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M285">View MathML</a>

is clearly bounded because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M286">View MathML</a> for some positive constant C. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M287">View MathML</a>. Define the superposition operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M288">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M289">View MathML</a>

If I is a bounded interval in ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M291">View MathML</a> are chosen from (F2), then Φ is bounded because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M292">View MathML</a>

Since Y is a generalized ideal space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M293">View MathML</a> is a regular ideal space (since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M293">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M295">View MathML</a>-condition), Theorem 6.4 of [34] implies that Φ is continuous on Y. Recalling the fact that the conjugate function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M296">View MathML</a> is strictly less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M297">View MathML</a>, we know by Lemma 2.6 that the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M298">View MathML</a> is continuous and compact and so is the adjoint operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M299">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M300">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M114">View MathML</a>. From the relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M302">View MathML</a>, it follows that F is continuous and compact. In particular, if we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M303">View MathML</a>, then G is continuous and compact. This completes the proof. □

We observe the behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M304">View MathML</a> at infinity.

Lemma 3.6Under assumptions (F1) and (F3), the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M305">View MathML</a>has the following property:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M306">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M307">View MathML</a> be arbitrary. Choose a positive constant R such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M308">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M309">View MathML</a>. Since b is locally bounded, there is a nonnegative constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M310">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M311">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M312">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M250">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M314">View MathML</a>. By assumption (F3), Minkowski’s inequality and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M315">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M316">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M250">View MathML</a>, where C are some positive constants. It follows from Hölder’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M318">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M319">View MathML</a>. Therefore, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M320">View MathML</a>

 □

Recall that a real number μ is called an eigenvalue of (E) if the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M321">View MathML</a>

has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M322">View MathML</a> in X that is different from the origin.

Now we consider the following spectral result for nonhomogeneous operators. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M1">View MathML</a> is a constant function, the following assertion has been shown to hold by virtue of the Furi-Martelli-Vignoli spectrum; see Theorem 4 of [35] or Lemma 27 of [10].

Lemma 3.7Ifμis not an eigenvalue of (E), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M324">View MathML</a>

Proof Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M325">View MathML</a>

Choose an unbounded sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M216">View MathML</a> in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M327">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M328">View MathML</a>

(3.21)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M329">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M330">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M331">View MathML</a>

Hence it follows from Proposition 3.4 and (3.21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M332">View MathML</a>

(3.22)

By the compactness of G, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M333">View MathML</a> converges to some point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M334">View MathML</a>. From (3.22) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M335">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M215">View MathML</a>. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M337">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M338">View MathML</a> is a homeomorphism (see Theorem 3.2 in [18]), we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M339">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M340">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M215">View MathML</a> and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M342">View MathML</a>

We conclude that μ is an eigenvalue of (E). This completes the proof. □

4 Main result

In this section, we are preparing to prove our main result. First we give the definition of weak solutions for our problem.

Definition 4.1 A weak solution of (B) is a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M343">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M344">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M345">View MathML</a>

where J, F and G are defined by (3.3), (3.19) and (3.20), respectively.

The following result about the existence of an unbounded branch of solutions for nonlinear operator equations is taken from Theorem 2.2 of [11] (see also [10]) as a key tool in obtaining our bifurcation result.

Lemma 4.2LetXbe a Banach space andYbe a normed space. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M346">View MathML</a>is a homeomorphism and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M347">View MathML</a>is a continuous and compact operator such that the composition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M348">View MathML</a>is odd. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M349">View MathML</a>be a continuous and compact operator. If the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M350">View MathML</a>

is bounded, then the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M351">View MathML</a>

has an unbounded connected set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M352">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M353">View MathML</a>intersects<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M354">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M348">View MathML</a> is odd, Borsuk’s theorem implies that the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M356">View MathML</a>

is satisfied for all sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M357">View MathML</a>, where I is the identity operator on X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M358">View MathML</a> is the open ball in X centered at 0 of radius r, respectively. In view of Theorem 2.2 of [11], the conclusion holds. □

Based on the above lemma, we now can prove the main result on bifurcation result for problem (B).

Theorem 4.3Suppose that conditions (HJ1)-(HJ4) and (F1)-(F3) are satisfied. Ifμis not an eigenvalue of (E), then there is an unbounded connected set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M352">View MathML</a>such that every point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M343">View MathML</a>inCis a weak solution of the above problem (B) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M353">View MathML</a>intersects<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M354">View MathML</a>.

Proof Apply Lemma 4.2 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M363">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M364">View MathML</a>. From Lemmas 3.3 and 3.5 we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M141">View MathML</a> is a homeomorphism, the operators G and F are continuous and compact, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M366">View MathML</a> is odd. Since μ is not an eigenvalue of (E), Lemmas 3.6 and 3.7 imply that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M367">View MathML</a>, there is a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M368">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M369">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M250">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M371">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M372">View MathML</a>. Therefore, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M373">View MathML</a>

is bounded. By Lemma 4.2, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M374">View MathML</a>

contains an unbounded connected set C which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M353">View MathML</a> intersects <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M354">View MathML</a>. This completes the proof. □

Finally, we give an example which illustrates an application of our bifurcation result.

Example 4.4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M377">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M378">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M379">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M380">View MathML</a> and there is a real number δ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M381">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M382">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M384">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M385">View MathML</a>. If μ is not an eigenvalue of (E) and assumptions (F1)-(F3) are fulfilled, then there is an unbounded connected set C intersecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M354">View MathML</a> such that every point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M343">View MathML</a> in C is a weak solution of the nonlinear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M388">View MathML</a>

Proof Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M389">View MathML</a>, we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M390">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M391">View MathML</a>, the inequality is clear. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M339">View MathML</a>. It follows from Young’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M393">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M394">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M395">View MathML</a> and choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M396">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M397">View MathML</a> for almost everywhere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M399">View MathML</a>

Thus (HJ1) and (HJ2) are satisfied. Removing some null set from Ω if necessary, we may suppose that the hypotheses are satisfied for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>. If we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M401">View MathML</a>

we observe that the first relation in (3.1) holds, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M402">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M403">View MathML</a>. Moreover, a straightforward calculation shows that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M404">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M405">View MathML</a>

From an analogous argument in the proof of Corollary 3.2 in [11], we can show that this expression is bounded from below by a positive constant which is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M406">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M96">View MathML</a>. Therefore (HJ3) is satisfied. Finally, (HJ4) holds if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M408">View MathML</a> we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M409">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/27/mathml/M64">View MathML</a>. □

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This research was supported by a 2011 Research Grant from Sangmyung University.

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