Open Access Research

Existence and multiplicity of solutions for nonlocal p ( x ) -Laplacian problems in R N

Erlin Guo* and Peihao Zhao

Author Affiliations

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China

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Boundary Value Problems 2012, 2012:79  doi:10.1186/1687-2770-2012-79


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/79


Received:13 March 2012
Accepted:9 July 2012
Published:26 July 2012

© 2012 Guo and Zhao; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we study the nonlocal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian problem of the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M4">View MathML</a>

By using the method of weight function and the theory of the variable exponent Sobolev space, under appropriate assumptions on f and M, we obtain some results on the existence and multiplicity of solutions of this problem. Moreover, we get much better results with f in a special form.

MSC: 35B38, 35D05, 35J20.

Keywords:
critical points; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian; nonlocal problem; variable exponent Sobolev spaces

1 Introduction

In this paper, we consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M6">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a> is a function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M9">View MathML</a> is a continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M10">View MathML</a> satisfies the Caratheodory condition.

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M11">View MathML</a> is called <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian, which becomes p-Laplacian when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M13">View MathML</a> (a constant). The <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian possesses more complicated nonlinearities than p-Laplacian; for example, p-Laplacian is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M15">View MathML</a>-homogeneous, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M16">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M17">View MathML</a>; but the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian operator, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a> is not a constant, is not homogeneous. These problems with variable exponent are interesting in applications and raise many difficult mathematical problems. Some of the models leading to these problems of this type are the models of motion of electrorheological fluids, the mathematical models of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium. We refer the reader to [1-7] for the study of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian equations and the corresponding variational problems.

Kirchhoff has investigated the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M21">View MathML</a>

which is called the Kirchhoff equation. This equation is an extension of the classical d’Alembert’s wave equation by considering the effect of the changes in the length of the string during vibrations. A distinguishing feature of the Kirchhoff equation is that the equation contains a nonlocal coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M22">View MathML</a> which depends on the average <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M23">View MathML</a> of the kinetic energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M24">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M25">View MathML</a>. Various equations of Kirchhoff type have been studied by many authors, especially after the work of Lions [8], where a functional analysis framework for the problem was proposed; see, e.g., [9-24] for some interesting results and further references. And now the study of a nonlocal elliptic problem has already been extended to the case involving the p-Laplacian; see, e.g., [25,26]. Corrêa and Figueiredo in [16] present several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the p-Kirchhoff type equation. Recently, the Kirchhoff type equation involving the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M27">View MathML</a>

has been investigated by Autuori, Pucci and Salvatori [27]. In [28] Fan studied <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Kirchhoff type equations with Dirichlet boundary value problems. Many papers are about these problems in bounded domains. According to the information I have, for Kirchhoff-type problems in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2">View MathML</a>, the results are seldom, in [29] Jin and Wu obtained three existence results of infinitely many radial solutions for Kirchhoff-type problems in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2">View MathML</a>, and in [30] Ji established the existence of infinitely many radially symmetric solutions of Kirchhoff-type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M31">View MathML</a>-Laplacian equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M32">View MathML</a>. The main difficulty here arises from the lack of compactness. Jin [29] and Ji [30] investigated these problems in radial symmetric spaces. In this paper, to deal with problem (P), we overcome the difficulty caused by the absence of compactness through the method of weight function. We establish conditions ensuring the existence and multiplicity of solutions for the problem.

This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we obtain the solutions with negative energy by the coercivity of functionals, and in Section 4, we obtain the solutions with positive energy by the Mountain Pass Theorem. Finally in Section 5, we obtain the infinity of solutions by the Fountain Theorem and the Dual Fountain Theorem when f satisfies a special form.

2 Preliminaries

In order to discuss problem (P), we need some theories on space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33">View MathML</a> which we call variable exponent Sobolev space. Firstly, we state some basic properties of space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33">View MathML</a> which will be used later (for details, see [6,31,32]).

Let Ω be an open domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M36">View MathML</a> the set of all measurable real functions defined on Ω, elements in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M36">View MathML</a> which are equal to each other and almost everywhere are considered as one element, and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M38">View MathML</a>

we can introduce the norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M39">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M40">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M41">View MathML</a> becomes a Banach space. We call it a variable exponent Lebesgue space.

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M43">View MathML</a>

and it can be equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M44">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M45">View MathML</a>; and we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M46">View MathML</a> the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M47">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M50">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M51">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M52">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M53">View MathML</a>.

Proposition 2.1 (see [6] and [31])

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M54">View MathML</a>, the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M41">View MathML</a>is a separable, uniform convex Banach space, and its dual space is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M56">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M57">View MathML</a>. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M59">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M60">View MathML</a>

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M61">View MathML</a>, then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M63">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M65">View MathML</a>

Proposition 2.2 (see [6])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M66">View MathML</a>is a Caratheodory function and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M67">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M71">View MathML</a>is a constant, then the superposition operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M72">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M73">View MathML</a>defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M74">View MathML</a>is a continuous and bounded operator.

Proposition 2.3 (see [6])

If we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M75">View MathML</a>

then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M76">View MathML</a>

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M77">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M78">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M79">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M80">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M81">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M82">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M81">View MathML</a>.

Proposition 2.4 (see [6])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M85">View MathML</a>, then the following statements are equivalent to each other

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M86">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M87">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M88">View MathML</a>in measure in Ω and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M89">View MathML</a>.

Proposition 2.5 (see [6])

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M90">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M33">View MathML</a>are separable reflexive Banach spaces.

Proposition 2.6If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M93">View MathML</a>is Lipschitz continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M94">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M95">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M96">View MathML</a>, there is a continuous embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M97">View MathML</a>.

For any measurable functions α, β, use the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M98">View MathML</a> to denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M99">View MathML</a>

Proposition 2.7Let Ω be a bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M94">View MathML</a>. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M103">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M104">View MathML</a>, there is a compact embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M97">View MathML</a>.

Proposition 2.8 (Poincare inequality)

There is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M106">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M107">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M108">View MathML</a>is a norm equivalent to the norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M109">View MathML</a>in the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M46">View MathML</a>.

3 Solutions with negative energy

In the following sections, we consider problem (P), the nonlocal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M1">View MathML</a>-Laplacian problem with variational form, where M is a real function satisfying the following condition: (M1) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M112">View MathML</a> is continuous and bounded.. And we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M114">View MathML</a> is Lipschitz continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M116">View MathML</a> satisfies Caratheodory conditions.

For simplicity, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M117">View MathML</a>. Denote by C a general positive constant (the exact value may change from line to line).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M119">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M120">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M121">View MathML</a>.

Before giving our main results, we first give several lemmas that will be used later.

Lemma 3.1 (see [2] and [28])

Let (M1) hold. Then the following statements hold:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M124">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M125">View MathML</a>.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M128">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M129">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M130">View MathML</a>.

Lemma 3.2 (see [2])

Suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M131">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M136">View MathML</a>, and there are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M137">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M138">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M139">View MathML</a>and Φ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M140">View MathML</a>are weakly-strongly continuous, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M141">View MathML</a>implies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M142">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M143">View MathML</a>.

Lemma 3.3

(1) The functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M144">View MathML</a>is sequentially weakly lower semi-continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M145">View MathML</a>is sequentially weakly continuous, and thusEis sequentially weakly lower semi-continuous.

(2) For any open set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M146">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M147">View MathML</a>, the mappings<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M148">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M149">View MathML</a>are bounded, and are of type<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M150">View MathML</a>, namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M151">View MathML</a>

Proof Since the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M152">View MathML</a> is increasing and the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M153">View MathML</a> is sequentially weakly lower semi-continuous, we conclude that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M144">View MathML</a> is sequentially weakly lower semi-continuous. From Lemma 3.2, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M155">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M156">View MathML</a> are sequentially weakly-strongly continuous. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M147">View MathML</a>. It is clear that the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M159">View MathML</a> are bounded. To prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M160">View MathML</a> is of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M150">View MathML</a>, assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M141">View MathML</a> in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M164">View MathML</a>, then there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M166">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M167">View MathML</a>. Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M168">View MathML</a>. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M169">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M170">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M171">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M172">View MathML</a> is of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M150">View MathML</a>. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M156">View MathML</a> is sequentially weakly-strongly continuous, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M159">View MathML</a> is of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M150">View MathML</a>. □

Definition 3.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M177">View MathML</a>. A <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M178">View MathML</a>-functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M179">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180">View MathML</a> condition if and only if every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M181">View MathML</a> in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M182">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M183">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M184">View MathML</a> has a convergent subsequence.

Lemma 3.4 (see [28])

Supposefsatisfies the hypotheses in Lemma 3.2, and let (M1) hold. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M185">View MathML</a>, every bounded<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180">View MathML</a>sequence forE, i.e., a bounded sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M187">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M188">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M189">View MathML</a>, has a strongly convergent subsequence.

As X is a separable and reflexive Banach space, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M190">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M191">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M192">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193">View MathML</a> , denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M194">View MathML</a>

Lemma 3.5 (see [2])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M195">View MathML</a>is weakly-strongly continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M197">View MathML</a>is a given positive number. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M198">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M199">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>.

Theorem 3.1Supposefsatisfies the hypotheses in Lemma 3.2, let (M1) hold and the following conditions hold: (M2) = There are positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M201">View MathML</a>, MandCsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M202">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M203">View MathML</a>.; (H1) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M204">View MathML</a>..Then the functionalEis coercive and attains its infimum inXat some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M205">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206">View MathML</a>is a solution of (P) ifEis differentiable at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206">View MathML</a>, and in particular, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M208">View MathML</a>.

Proof We have concluded that E is weakly lower semi-continuous. Let us prove that E is coercive on X, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M209">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M210">View MathML</a>. For simplicity, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M211">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M215">View MathML</a>. We have that

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M109">View MathML</a> is large enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M218">View MathML</a>

and hence E is coercive. Since E is sequentially weakly lower semi-continuous and X is reflexive, E attains its infimum in X at some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M205">View MathML</a>. In the case where E is differentiable at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206">View MathML</a> is a solution of (P). □

Theorem 3.2Supposefsatisfies the hypotheses in Lemma 3.2. Let (M1), (M2), (H1) and the following conditions hold: (M3) = There is a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M222">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M223">View MathML</a>.; (f1) = There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M224">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M225">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M230">View MathML</a>.; (H2) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M231">View MathML</a>..Then (P) has at least one nontrivial solution which is a global minimizer of the energy functionalE.

Proof From Theorem 3.1 we know that E has a global minimizer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M233">View MathML</a> and consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M234">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M228">View MathML</a>, we can find a bounded open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M237">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M238">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M239">View MathML</a>. The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M240">View MathML</a> is a subspace of X. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M241">View MathML</a>. Then, by (f1), (M3) and (H2), for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M17">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M243">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M244">View MathML</a> which shows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M208">View MathML</a>. □

Theorem 3.3Let the hypotheses of Theorem 3.2 hold, andfsatisfy the following condition: (f2) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M246">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M247">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M248">View MathML</a>..Then (P) has a sequence of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M249">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M250">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M251">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>.

Proof Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M253">View MathML</a> the genus of A. Denote

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M255">View MathML</a>.

From the condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M256">View MathML</a>, there exists a bounded open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M257">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M238">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M239">View MathML</a>. The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M260">View MathML</a> is a subspace of X. For any k, we can choose a k-dimensional linear subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M261">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M260">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M263">View MathML</a>. As the norms on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M261">View MathML</a> are equivalent to each other, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M265">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M266">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M267">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M268">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M269">View MathML</a> is compact, and then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M270">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M271">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M272">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M273">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M274">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M231">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M276">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M277">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M279">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M281">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M282">View MathML</a>, we get the conclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M283">View MathML</a>.

By the genus theory, each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M284">View MathML</a> is a critical value of E, hence there is a sequence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M285">View MathML</a> of problem (P) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M286">View MathML</a>.

At last, we will prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M287">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>. By the coercive of E, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M197">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M290">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M291">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M292">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M293">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M294">View MathML</a> be the subspace of X as mentioned above. According to the properties of genus, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M295">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M296">View MathML</a>

we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M199">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M299">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M300">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M301">View MathML</a>, and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M302">View MathML</a>, which concludes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M287">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>. □

Theorem 3.4Let the hypotheses of Lemma 3.2, (f1), (M1), (M2), (M3), (H1), (H2) and the following condition hold, (f+) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M305">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M247">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M118">View MathML</a>..Then (P) has at least one nontrivial nonnegative solution with negative energy.

Proof Define

Then, like in the proof of Theorem 3.2, using truncation functions above, similarly to the proof of Theorem 3.4 in [28], we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M309">View MathML</a> has a nontrivial global minimizer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M206">View MathML</a> is a nontrivial nonnegative solution of (P). □

4 Solution with positive energy

In this section we will find the Mountain Pass type critical points of the energy functional E associated with problem (P).

Lemma 4.1Let (f1), (M1) and the following conditions hold: (M2) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M313">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M106">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M315">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M316">View MathML</a>hold.; (M4) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M317">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M313">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M319">View MathML</a>

; (f3) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M313">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M322">View MathML</a>

; (H3) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M323">View MathML</a>..Then E satisfies condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M185">View MathML</a>.

Proof By (M4), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M109">View MathML</a> large enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M327">View MathML</a>

By (f3) we conclude that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M328">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M329">View MathML</a>

and thus, given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M330">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M331">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M332">View MathML</a>

we claim that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M333">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M334">View MathML</a>

the notation of this conclusion can be seen in [28].

Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M189">View MathML</a>. By (H3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M338">View MathML</a> small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M339">View MathML</a>. Then, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M340">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180">View MathML</a> sequence, for sufficiently large n, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M342">View MathML</a>

we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M343">View MathML</a> is bounded, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M316">View MathML</a>. By Lemma 3.4, E satisfies condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M180">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M185">View MathML</a>. □

Lemma 4.2Under the hypotheses of Lemma 4.1, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M347">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M348">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M349">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M347">View MathML</a> be given. From (M4) for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M125">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M352">View MathML</a>

and then it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M353">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M354">View MathML</a> is a positive constant depending on w. From (f4) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M355">View MathML</a> large enough we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M356">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M357">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M358">View MathML</a> is a positive constant depending on w. Hence for s large enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M359">View MathML</a>

and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M348">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M361">View MathML</a>. □

Lemma 4.3Under the hypotheses of Lemma 3.2, (M1) holds and the following conditions hold: (M5) = There is a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M362">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M363">View MathML</a>.; (f4) = There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M364">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M365">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M247">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M367">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M247">View MathML</a>.; (H4) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M369">View MathML</a>..

Then there exist positive constantsρandδsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M370">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M371">View MathML</a>.

Proof It follows from (M5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M372">View MathML</a>

It follows from the hypotheses of Lemma 3.2 and (f4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M373">View MathML</a>

Thus by (H4), we obtain the assertion of Lemma 4.3. □

By the famous Mountain Pass lemma, from Lemmas 4.1-4.3, we have the following:

Theorem 4.1Let all hypotheses of Lemmas 4.1-4.3 hold. Then (P) has a nontrivial solution with positive energy.

5 The case of concave-convex nonlinearity

In this section, we will obtain much better results with f in a special form. We have the following theorem:

Theorem 5.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M374">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M375">View MathML</a>

Then we have

(1) If (M1), (M2), (M4), (H3) hold and we also assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M376">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M377">View MathML</a>, then problem (P) has solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M378">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M379">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>.

(2) If (M1), (M4), (M5), (H3) hold and we also assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M381">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M382">View MathML</a>, then problem (P) has solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M383">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M384">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M385">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>.

We will use the following ‘Fountain Theorem’ and the ‘Dual Fountain Theorem’ to prove Theorem 5.1.

Proposition 5.1 (Fountain Theorem, see [11])

Assume(A1) = Xis a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M387">View MathML</a>is an even functional, the subspaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M388">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M293">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M294">View MathML</a>are defined by (3.2)..

If for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M392">View MathML</a>such that(A2) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M393">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>.; (A3) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M395">View MathML</a>.; (A4) = Esatisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M396">View MathML</a>condition for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M397">View MathML</a>. ThenEhas a sequence of critical values tending to +∞..

Proposition 5.2 (Dual Fountain Theorem, see [11])

Assume (A1) is satisfied and there is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M398">View MathML</a>so as to for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M399">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M392">View MathML</a>such that(B1) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M401">View MathML</a>.; (B2) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M402">View MathML</a>.; (B3) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M403">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>.; (B4) = Esatisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M405">View MathML</a>condition for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M406">View MathML</a>. ThenEhas a sequence of negative critical values converging to 0..

Definition 5.1 We say that E satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M405">View MathML</a> condition (with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M408">View MathML</a>), if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M409">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M410">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M411">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M412">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M413">View MathML</a>, contains a subsequence converging to a critical point of E.

Proof of Theorem 5.1 Firstly, we verify the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M405">View MathML</a> condition for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M177">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M410">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M412">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M419">View MathML</a>. It is easy to obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M420">View MathML</a> satisfies condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M421">View MathML</a>), when it has this special form. So similar to the method in Lemma 4.1, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M422">View MathML</a>

hence, we can get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M423">View MathML</a> is bounded. Going if necessary to a subspace, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M424">View MathML</a> in X. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M425">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M426">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M427">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M428">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M429">View MathML</a> is of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M430">View MathML</a> type, we can conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M431">View MathML</a>; furthermore, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M432">View MathML</a>.

It only remains to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M433">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M434">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M435">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M436">View MathML</a>

Going to the limit on the right side of the above equation reaches

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M437">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M433">View MathML</a>, this shows that E satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M405">View MathML</a> condition for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M177">View MathML</a>. Obviously, E also satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M396">View MathML</a> condition for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M177">View MathML</a>.

(1) We will prove that if k is large enough, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M392">View MathML</a> such that (A2) and (A3) are satisfied. (A2) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193">View MathML</a> , denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M445">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M446">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M447">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M448">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M199">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M200">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M299">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M452">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M453">View MathML</a>

For sufficiently large k, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M454">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M376">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M456">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M457">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M458">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M199">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M460">View MathML</a>. (A2) is satisfied.

(A3) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193">View MathML</a> , denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M462">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M463">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M464">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465">View MathML</a> and t large enough, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M466">View MathML</a>, all norms are equivalent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M293">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M468">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M469">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M470">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M471">View MathML</a> concludes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M472">View MathML</a> and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M473">View MathML</a>

so (A2) is satisfied.

Conclusion (1) is reached by the Fountain Theorem.

(2) We use the Dual Fountain Theorem to prove conclusion (2), and now it remains for us to prove that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M392">View MathML</a> such that if k is large enough (B1), (B2) and (B3) are satisfied.

(B1) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M475">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M476">View MathML</a> be defined as above, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M477">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465">View MathML</a> and t small enough we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M479">View MathML</a>

For sufficiently large k we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M480">View MathML</a>, thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M481">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M482">View MathML</a>, then for sufficiently large k, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M483">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M471">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M477">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M487">View MathML</a>, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M488">View MathML</a>

Hence (B1) is satisfied.

(B2) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M193">View MathML</a> , denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M490">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M491">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M464">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465">View MathML</a> and t small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M494">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M466">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M382">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M497">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M498">View MathML</a> small enough. Hence (B2) is satisfied.

(B3) From the proof above and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M499">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M500">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M477">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M465">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M503">View MathML</a> small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M504">View MathML</a>

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/79/mathml/M505">View MathML</a>. Hence (B3) is satisfied.

Conclusion (2) is reached by the Dual Fountain Theorem. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

EG and PZ contributed to each part of this work equally. All the authors read and approved the final manuscript.

Acknowledgements

The authors thank the two referees for their careful reading and helpful comments on the study. Research was supported by the National Natural Science Foundation of China (10971088), (10971087) and the Fundamental Research Funds for the Central Universities (lzujbky-2012-180).

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