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Existence of an unbounded branch of the set of solutions for Neumann problems involving the p(x)-Laplacian

Byung-Hoon Hwang12, Seung Dae Lee2 and Yun-Ho Kim2*

Author Affiliations

1 Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea

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

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Boundary Value Problems 2014, 2014:92  doi:10.1186/1687-2770-2014-92

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/92


Received:11 February 2014
Accepted:11 April 2014
Published:6 May 2014

© 2014 Hwang et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

We are concerned with the following nonlinear problem: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M2">View MathML</a> in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M3">View MathML</a> on Ω, which is subject to a Neumann boundary condition, provided that μ is not an eigenvalue of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian. The aim of this paper is to study the structure of the set of solutions for the degenerate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian Neumann problems by applying a bifurcation result for nonlinear operator equations.

MSC: 35B32, 35D30, 35J70, 47J10, 47J15.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian; weighted variable exponent Lebesgue-Sobolev spaces; Neumann boundary condition; eigenvalue

1 Introduction

In recent years, there has been much interest in studying differential equations and variational problems involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-growth conditions 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 [1-5] and references therein. In the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a> a constant, called the p-Laplacian, there are a lot of papers, for instance, [6-14] and references therein.

In the present paper, we are concerned with the existence of an unbounded branch of the set of solutions for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian problem with degeneracy subject to the Neumann boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M10">View MathML</a>

(B)

when μ is not an eigenvalue of the divergence form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M11">View MathML</a>

(E)

where Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M12">View MathML</a> with the Lipschitz boundary Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M13">View MathML</a> denotes the outer normal derivative of u with respect to Ω, the variable exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M14">View MathML</a> is a continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M15">View MathML</a>, w is a weighted function in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M16">View MathML</a> satisfies a Carathéodory condition.

Since the inceptive study of bifurcation theory by Krasnoselskii [15], Rabinowitz [16] claimed that the bifurcation occurring in the Krasnoselskii theorem is actually a global phenomenon. As regards the p-Laplacian and generalized operators, the nonlinear eigenvalue and bifurcation problems have been widely studied by many researchers in various approaches in the spirit of Rabinowitz [16]; see also [6-9,13,17].

The authors in [6,7] obtained the bifurcation phenomenon for the nonlinear Dirichlet problem which bifurcates from the first eigenvalue of the p-Laplacian. As in [6,7], Khalil and Ouanan [18] got the result for the nonlinear Neumann problem of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M17">View MathML</a>

(A)

which is based on the fact [19] that the first eigenvalue of the p-Laplacian is simple and isolated under suitable conditions on m.

While many researchers considered global branches bifurcating from the first eigenvalue of the p-Laplacian, Väth [20] came at it from another viewpoint to establish the existence of a global branch of solutions for the p-Laplacian with Dirichlet boundary condition by applying nonlinear spectral theory for homogeneous operators. From this point of view, for the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a> is a constant function, the existence of a global branch of solutions for the problem (B) was attained in [11] (for generalization to equations involving nonhomogeneous operators, see also [12]) when μ is not eigenvalue of (E).

Compared to the p-Laplacian equation, an analysis for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian equation has to be carried out more carefully because it has complicated nonlinearities (it is nonhomogeneous) and includes a weighted function. As mentioned before, the fact that the principal eigenvalue for nonlinear eigenvalue problems related to the p-Laplacian under either Dirichlet boundary condition or Neumann boundary condition is isolated plays a key role in obtaining the bifurcation result from the principal eigenvalue of the p-Laplacian. However, unlike the p-Laplacian case, under some conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>, the first eigenvalue for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian Neumann problems is not isolated (see [21]), that is, the infimum of all eigenvalues of the problem might be zero (see [22] for Dirichlet boundary condition). Thus we cannot investigate the existence of global branches bifurcating from the principal eigenvalue of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian. For this reason, the global behavior of solutions for nonlinear problems involving the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian had been considered in [23]. To the best of our knowledge, there are no papers concerned with the bifurcation theory for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian Neumann problems with weighted functions.

This paper is organized as follows. We first state some basic results for the weighted variable exponent Lebesgue-Sobolev spaces which were given in [23]. Next we give some properties of the corresponding integral operators. Finally we show the existence of a global bifurcation for a Neumann problem involving the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian by using a bifurcation result in an abstract setting.

2 Preliminaries

In this section, we state some elementary properties for the (weighted) variable exponent Lebesgue-Sobolev spaces which will be used in the next sections. The basic properties of the variable exponent Lebesgue-Sobolev spaces, that is, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M26">View MathML</a> can be found from [24].

To make a self-contained paper, we recall some definitions and basic properties of the weighted variable exponent Lebesgue spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M27">View MathML</a> and the weighted variable exponent Lebesgue-Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M28">View MathML</a>.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M29">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M30">View MathML</a> we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M31">View MathML</a>

Let w is a measurable positive and a.e. finite function in Ω. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M32">View MathML</a>, we introduce the weighted variable exponent Lebesgue space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M33">View MathML</a>

endowed with the Luxemburg norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M34">View MathML</a>

The weighted variable exponent Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M35">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M36">View MathML</a>

where the norm is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M37">View MathML</a>

(2.1)

It is significant that smooth functions are not dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M38">View MathML</a> without additional assumptions on the exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>. This feature was observed by Zhikov [25] in connection with the Lavrentiev phenomenon. However, if the exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a> is log-Hölder continuous, i.e., there is a constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M41">View MathML</a>

(2.2)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M42">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M43">View MathML</a>, then smooth functions are dense in variable exponent Sobolev spaces and there is no confusion in defining the Sobolev space with zero boundary values, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M44">View MathML</a>, as the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M45">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M46">View MathML</a> (see [26]).

Lemma 2.1 ([24])

The space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M47">View MathML</a>is a separable, uniformly convex Banach space, and its conjugate space is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M48">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M49">View MathML</a>. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M50">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M51">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M52">View MathML</a>

Lemma 2.2 ([23])

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M53">View MathML</a>

Then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M54">View MathML</a> (=1; <1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M55">View MathML</a> (=1; <1), respectively;

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M56">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M57">View MathML</a>;

(3) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M58">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M59">View MathML</a>.

Lemma 2.3 ([27])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M60">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M61">View MathML</a>for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M63">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M64">View MathML</a>, then

(1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M65">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M66">View MathML</a>;

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M67">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M68">View MathML</a>.

We assume that w is a measurable positive and a.e. finite function in Ω satisfying that

(w1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M70">View MathML</a>;

(w2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M71">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M72">View MathML</a>.

The reasons that we assume (w1) and (w2) can be found in [23].

Lemma 2.4 ([23])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M32">View MathML</a>and (w1) hold. ThenXis a reflexive and separable Banach space.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M74">View MathML</a>, let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M75">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M76">View MathML</a> is given in (w2) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M77">View MathML</a>

(2.3)

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a>.

We shall frequently make use of the following (compact) imbedding theorem for the weighted variable exponent Lebesgue-Sobolev space in the next sections.

Lemma 2.5 ([23])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M79">View MathML</a>be an open, bounded set with Lipschitz boundary and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M32">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M81">View MathML</a>satisfy the log-Hölder continuity condition (2.2). If assumptions (w1) and (w2) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M82">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M83">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M84">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M86">View MathML</a>

and the imbedding is compact if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M87">View MathML</a>.

3 Properties of the integral operators

In this section, we give the definitions and some properties of the integral operators corresponding to the problem (B), by applying the basic properties of the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M88">View MathML</a> and X which are given in the previous section.

Throughout this paper, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M32">View MathML</a> satisfy the log-Hölder continuity condition (2.2). We define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M90">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M91">View MathML</a>

(3.1)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M92">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M93">View MathML</a> denotes the pairing of X and its dual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94">View MathML</a> and the Euclidean scalar product on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M12">View MathML</a>, respectively.

The following estimate, which can be found in [17], plays a key role in obtaining the homeomorphism of the operator J.

Lemma 3.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M96">View MathML</a>, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M97">View MathML</a>

From Lemma 3.1, we can obtain the following topological result, which will be needed in the main result. Compared to the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a> being constant (see [11]), the following result is hard to prove because it has complicated nonlinearities.

Theorem 3.2Let (w1) and (w2) be satisfied. The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M99">View MathML</a>is homeomorphism onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94">View MathML</a>with a bounded inverse.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M102">View MathML</a> be operators defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M103">View MathML</a>

Then the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M105">View MathML</a> are bounded and continuous. In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M106">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M107">View MathML</a> in X as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M108">View MathML</a>. Then there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M109">View MathML</a> and functions v, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M110">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M88">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M112">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M113">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M116">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M117">View MathML</a> and for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a>. Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M119">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M120">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M121">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M122">View MathML</a>

(3.3)

and the integrands at the right-hand sides in (3.2) and (3.3) are dominated by some integrable functions. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M123">View MathML</a> in X as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M114">View MathML</a>, we can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M126">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M114">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M128">View MathML</a>. Therefore, the Lebesgue dominated convergence theorem tells us that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M129">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M131">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M132">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M114">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M105">View MathML</a> are continuous on X. Also it is easy to show that these operators are bounded on X.

Using the continuity for the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M105">View MathML</a> on X, we finally show that J is continuous on X. From Hölder’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M138">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M92">View MathML</a>. Hence we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M140">View MathML</a>

(3.4)

and the right-hand side in (3.4) converges to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M141">View MathML</a>. Therefore the operator J is continuous on X.

For any u in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M142">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M143">View MathML</a>

for some positive constant C. Thus we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M144">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M145">View MathML</a> and therefore the operator J is coercive on X.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M146">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M147">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M148">View MathML</a>

(Of course, if both the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M150">View MathML</a> are nonempty, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M151">View MathML</a> by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>.) It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M153">View MathML</a>

(3.5)

By using Lemma 3.1 and (3.5), we find that J is strictly monotone on X. The Browder-Minty theorem hence implies that the inverse operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M154">View MathML</a> exists and is bounded; see Theorem 26.A in [28].

Next we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M155">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94">View MathML</a>. Assume that u and v are any elements in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M157">View MathML</a>. According to Lemma 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M158">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M159">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M160">View MathML</a> and for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M162">View MathML</a>. Integrating the above inequalities over Ω and using Lemma 2.2, we assert that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M163">View MathML</a>

(3.6)

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M165">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M166">View MathML</a>. For almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M167">View MathML</a>, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M168">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M169">View MathML</a>

(3.8)

where we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M170">View MathML</a> and use the shortcuts

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M171">View MathML</a>

Hence using Lemma 3.1, we assert that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M172">View MathML</a>

for some positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M173">View MathML</a>. From Hölder’s and Minkowski’s inequalities, and the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M174">View MathML</a>

(3.9)

for any positive numbers a, b, r, and s, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M175">View MathML</a>

(3.10)

Applying Lemma 2.3 and Minkowski’s inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M176">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M177">View MathML</a> where α is either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M178">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M179">View MathML</a>. In a similar way,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M180">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M177">View MathML</a> where β is either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M178">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M179">View MathML</a>. It follows from (3.7)-(3.10) and Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M184">View MathML</a>

where γ is either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M178">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M179">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M187">View MathML</a> is positive constant. So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M188">View MathML</a>

(3.11)

for some positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M189">View MathML</a>. Consequently, it follows from (3.6) and (3.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M190">View MathML</a>

(3.12)

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M192">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M193">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M194">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M195">View MathML</a> be any sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94">View MathML</a> that converges to h in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M199">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M200">View MathML</a>. We obtain from (3.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M201">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M202">View MathML</a> is bounded in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M203">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M94">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M205">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M206">View MathML</a> converges to u in X. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M155">View MathML</a> is continuous at each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M194">View MathML</a>. This completes the proof. □

From now on we deal with the properties for the superposition operator induced by the function f in (B). We assume that the variable exponents are subject to the following restrictions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M209">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a>. Assume that:

(F1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M211">View MathML</a> satisfies the Carathéodory condition in the sense that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M212">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M214">View MathML</a> is continuous for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a>.

(F2) For each bounded interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M216">View MathML</a>, there are a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M217">View MathML</a> and a nonnegative constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M218">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M219">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M221">View MathML</a>.

(F3) f satisfies the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M222">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M224">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M225">View MathML</a>.

(F4) There exist a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M226">View MathML</a> and a locally bounded function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M227">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M228">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M229">View MathML</a>

for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M231">View MathML</a>.

Under assumptions (F1) and (F2), we can define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M232">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M233">View MathML</a>

(3.13)

and an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M234">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M235">View MathML</a>

(3.14)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M92">View MathML</a>.

For our aim, we need some properties of the operators F and G. In contrast with [23], we give a direct proofs for the continuity and compactness of F and G without using a continuity result on superposition operators.

Theorem 3.3If (w1), (w2), and (F1)-(F3) hold, then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M237">View MathML</a>is continuous and compact. Also the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M234">View MathML</a>is continuous and compact.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M239">View MathML</a> be an operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M240">View MathML</a>

Then for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M241">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M242">View MathML</a> is bounded and continuous. In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M244">View MathML</a> in X as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M245">View MathML</a>. Then there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M109">View MathML</a> and functions v, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M110">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M248">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M249">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M250">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M251">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M252">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M254">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M117">View MathML</a> and for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M256">View MathML</a>. Suppose that we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M257">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M258">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M259">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M258">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M261">View MathML</a>

and (F2) implies that the integrand at the right-hand side is dominated by an integrable function. Since the function f satisfies a Carathéodory condition, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M262">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M252">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M62">View MathML</a>. Therefore, the Lebesgue dominated convergence theorem tells us that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M265">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M248">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M267">View MathML</a>. We conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M268">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M248">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M205">View MathML</a> and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M271">View MathML</a> is continuous on X. The boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M271">View MathML</a> follows from (F2), Minkowski’s inequality, and the imbedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M273">View MathML</a> continuously (see Theorem 2.11 in [23]) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M274">View MathML</a>

(3.15)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243">View MathML</a> and for some positive constant d.

Minkowski’s inequality and (3.12) imply in view of (F3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M276">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M277">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243">View MathML</a>. This shows that for any bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M279">View MathML</a>, the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M280">View MathML</a> is equicontinuous at each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M281">View MathML</a>. Hence it follows from the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M282">View MathML</a> that Ψ is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M283">View MathML</a>, on observing the following relation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M284">View MathML</a>

Moreover, Ψ is bounded. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M286">View MathML</a> are bounded, we have to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M287">View MathML</a> is bounded. We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M288">View MathML</a> is compact. By the equicontinuity and the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M288">View MathML</a>, we can find finitely many numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M290">View MathML</a> such that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M291">View MathML</a> there is an integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M292">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M293">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M294">View MathML</a> is bounded for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M292">View MathML</a>, Minkowski’s inequality hence implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M287">View MathML</a> is bounded.

Recall that the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M297">View MathML</a> is continuous and compact (see e.g.[8]) and so the adjoint operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M298">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M299">View MathML</a>

is also compact. As F can be written as a composition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M300">View MathML</a> with Ψ, we conclude that F is continuous and compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M283">View MathML</a>. The operator G is continuous and compact because G can be regarded as a special case of F. This completes the proof. □

The analog of the following result can be found in [23]. However, our growth condition described in assumption (F4) is slightly different from that of [23].

Lemma 3.4Let assumptions (w1), (w2), (F1) and (F4) be fulfilled. Then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M302">View MathML</a>has the following property:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M303">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M304">View MathML</a>. Choose a positive constant R such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M305">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M306">View MathML</a>. Since b is locally bounded, there is a nonnegative constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M307">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M308">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M309">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M142">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M312">View MathML</a>. Without loss of generality, we may suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M313">View MathML</a>

By assumption (F4), Lemma 2.5 and the continuous imbedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M314">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M315">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M316">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M317">View MathML</a> are positive constants. It follows from Hölder’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M318">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M319">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M142">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M321">View MathML</a> is a positive constant. Consequently, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M322">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/2014/1/92/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M323">View MathML</a>

has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M324">View MathML</a> in X which is different from the origin.

The following lemma is a consequence about nonlinear spectral theory and its proof can be found in [23]. For the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a> is a constant, this assertion has been obtained by using the Furi-Martelli-Vignoli spectrum; see Theorem 4 of [29] or Lemma 27 of [20].

Lemma 3.5Suppose that assumptions (w1) and (w2) are fulfilled. Ifμis not an eigenvalue of (E), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M326">View MathML</a>

(3.16)

4 Bifurcation result

In this section, we are ready to prove the main result. 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/2014/1/92/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M327">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M283">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M329">View MathML</a>

where J, F and G are defined by (3.1), (3.13) and (3.14), respectively.

The following result, taken from Theorem 2.2 of [20], is a key tool to obtain our bifurcation result.

Lemma 4.2LetXbe a Banach space andYbe a normed space. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M330">View MathML</a>is a homeomorphism and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M331">View MathML</a>is a continuous and compact operator such that the composition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M332">View MathML</a>is odd. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M333">View MathML</a>be a continuous and compact operator. If the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M334">View MathML</a>

is bounded, then the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M335">View MathML</a>

has an unbounded connected set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M336">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M337">View MathML</a>intersects<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M338">View MathML</a>.

Finally we establish the existence of an unbounded branch of the set of solutions for Neumann problem (B) thereby using Lemma 4.2.

Theorem 4.3Let conditions (w1), (w2), and (F1)-(F4) be satisfied. Ifμis not an eigenvalue of (E), then there is an unbounded connected set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M336">View MathML</a>such that every point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M327">View MathML</a>inCis a weak solution of the above problem (B) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M337">View MathML</a>intersects<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M338">View MathML</a>.

Proof By Theorem 3.2 and Lemma 3.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M343">View MathML</a> is a homeomorphism, the operators G and F are continuous and compact, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M344">View MathML</a> is odd. Since μ is not an eigenvalue of (E), we get by Lemma 3.5

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M345">View MathML</a>

This together with Lemma 3.4 implies that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M346">View MathML</a>, there is a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M347">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M348">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M243">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M350">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M351">View MathML</a>. Therefore, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M352">View MathML</a>

is bounded. By Lemma 4.2, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M353">View MathML</a>

contains an unbounded connected set C which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M337">View MathML</a> intersects <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M338">View MathML</a>. This completes the proof. □

In particular the following example illustrates an application of our bifurcation result.

Example 4.4 Suppose that assumptions (w1) and (w2) are fulfilled and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M15">View MathML</a>. If μ is not an eigenvalue of (E), then there is an unbounded connected set C such that every point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M327">View MathML</a> in C is a weak solution of the following nonlinear problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M358">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M359">View MathML</a> and the conjugate function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M360">View MathML</a> is strictly less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M361">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M362">View MathML</a>. Then it is clear that f satisfies conditions (F1)-(F4). Therefore, the conclusion follows from Theorem 4.3. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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