Existence of Solutions for the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq1_HTML.gif -Laplacian Problem with Singular Term

Boundary Value Problems20102010:584843

DOI: 10.1155/2010/584843

Received: 19 August 2009

Accepted: 7 March 2010

Published: 14 March 2010

Abstract

We study the following http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq2_HTML.gif -Laplacian problem with singular term: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq6_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq7_HTML.gif is a bounded domain, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq8_HTML.gif . We obtain the existence of solutions in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq9_HTML.gif .

1. Introduction

After Kováčik and Rákosník first discussed the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq10_HTML.gif spaces and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq11_HTML.gif spaces in [1], a lot of research has been done concerning these kinds of variable exponent spaces, for example, see [25] for the properties of such spaces and [69] for the applications of variable exponent spaces on partial differential equations. Especially in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq12_HTML.gif spaces, there are a lot of studies on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq13_HTML.gif -Laplacian problems; see [8, 9]. In the recent years, the theory of problems with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq14_HTML.gif -growth conditions has important applications in nonlinear elastic mechanics and electrorheological fluids (see [1014]).

In this paper, we study the existence of the weak solutions for the following http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq15_HTML.gif -Laplacian problem:

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq16_HTML.gif is a bounded domain, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq18_HTML.gif is Lipschitz continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq19_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq20_HTML.gif

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq21_HTML.gif be the set of all Lebesgue measurable functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq22_HTML.gif . For all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq23_HTML.gif , we denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq25_HTML.gif , and denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq26_HTML.gif the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq27_HTML.gif .

We impose the following condition on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq28_HTML.gif :

(F) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq31_HTML.gif and for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq32_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq33_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq34_HTML.gif , whenever http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq35_HTML.gif .

A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardy exponents of the form

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq38_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq39_HTML.gif . In fact, take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq41_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq42_HTML.gif , then it is easy to verify that (F) is satisfied.

Our object is to obtain the existence of solutions in the following four cases:

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq43_HTML.gif ;

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq44_HTML.gif ;

(3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq45_HTML.gif ;

(4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq46_HTML.gif .

When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq47_HTML.gif , the solution of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq48_HTML.gif -Laplacian equations without singularity has been studied by many researchers. The study of problem (1.1) with variable exponents is a new topic.

The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge of variable exponent Lebesgue and Sobolev spaces. In Section 3, we prove our main results.

2. Preliminaries

In this section we first recall some facts on variable exponent Lebesgue space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq49_HTML.gif and variable exponent Sobolev space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq50_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq51_HTML.gif is an open set; see [14, 8, 15] for the details.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq52_HTML.gif and

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ3_HTML.gif
(2.1)

The variable exponent Lebesgue space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq53_HTML.gif is the class of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq54_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq55_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq56_HTML.gif is a Banach space endowed with the norm (2.1).

For a given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq57_HTML.gif we define the conjugate function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq58_HTML.gif as

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ4_HTML.gif
(2.2)

Theorem 2.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq59_HTML.gif . Then the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ5_HTML.gif
(2.3)

holds for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq61_HTML.gif with the constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq62_HTML.gif depending on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq64_HTML.gif only.

Theorem 2.2.

The dual space of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq65_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq66_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq67_HTML.gif . The space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq68_HTML.gif is reflexive if and only if
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ6_HTML.gif
(2.4)

Theorem 2.3.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq69_HTML.gif satisfies (2.4). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq71_HTML.gif , then necessary and sufficient condition for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq72_HTML.gif is that for almost all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq73_HTML.gif one has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq74_HTML.gif , and in this case, the imbedding is continuous.

Theorem 2.4.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq75_HTML.gif satisfies (2.4). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq76_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq77_HTML.gif , then

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq78_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq79_HTML.gif ,

(2) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq80_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq81_HTML.gif ,

(3) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq82_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq83_HTML.gif ,

(4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq84_HTML.gif if and only if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq85_HTML.gif ,

(5) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq86_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq87_HTML.gif .

We assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq88_HTML.gif is a given positive integer.

Given a multi-index http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq89_HTML.gif , we set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq91_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq92_HTML.gif is the generalized derivative operator.

The generalized Sobolev space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq93_HTML.gif is the class of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq94_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq95_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq96_HTML.gif for every multi-index http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq97_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq98_HTML.gif , endowed with the norm

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ7_HTML.gif
(2.5)

By http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq99_HTML.gif we denote the subspace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq100_HTML.gif which is the closure of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq101_HTML.gif with respect to the norm (2.5).

In this paper we use the following equivalent norm on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq102_HTML.gif :

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ8_HTML.gif
(2.6)

Then we have the inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq103_HTML.gif .

Theorem 2.5.

The spaces http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq105_HTML.gif are separable reflexive Banach spaces, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq106_HTML.gif satisfies (2.4).

Theorem 2.6.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq107_HTML.gif satisfies (2.4). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq108_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq109_HTML.gif , then

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq110_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq111_HTML.gif ,

(2) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq112_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq113_HTML.gif ,

(3) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq114_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq115_HTML.gif ,

(4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq116_HTML.gif if and only if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq117_HTML.gif ,

(5) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq118_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq119_HTML.gif .

We denote the dual space of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq120_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq121_HTML.gif , then we have the following.

Theorem 2.7.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq122_HTML.gif . Then for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq123_HTML.gif there exists a unique system of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq124_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ9_HTML.gif
(2.7)
The norm of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq125_HTML.gif is defined as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ10_HTML.gif
(2.8)

Theorem 2.8.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq126_HTML.gif be a domain in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq127_HTML.gif with cone property. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq128_HTML.gif is Lipschitz continuous and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq130_HTML.gif is measurable and satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq131_HTML.gif a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq132_HTML.gif , then there is a continuous embedding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq133_HTML.gif .

Theorem 2.9.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq134_HTML.gif be a bounded domain. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq135_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq136_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ11_HTML.gif
(2.9)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq137_HTML.gif is a constant depending on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq138_HTML.gif .

Next let us consider the weighted variable exponent Lebesgue space. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq139_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq140_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq141_HTML.gif . Define

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ12_HTML.gif
(2.10)

with the norm

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ13_HTML.gif
(2.11)

then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq142_HTML.gif is a Banach space.

Theorem 2.10.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq143_HTML.gif satisfies (2.4). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq144_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq146_HTML.gif , then

(1) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq147_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq148_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq149_HTML.gif ,

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq150_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq151_HTML.gif ,

(3) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq152_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq153_HTML.gif ,

(4) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq154_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq155_HTML.gif ,

(5) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq156_HTML.gif if and only if    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq157_HTML.gif ,

(6) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq158_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq159_HTML.gif .

Theorem 2.11.

Assume that the boundary of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq160_HTML.gif possesses the cone property and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq161_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq162_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq163_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq164_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq165_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq166_HTML.gif then there is a compact embedding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq167_HTML.gif .

Theorem 2.12.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq168_HTML.gif be a measurable subset. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq169_HTML.gif be a Caracheodory function and satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ14_HTML.gif
(2.12)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq171_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq173_HTML.gif is a constant, then the Nemytsky operator from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq174_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq175_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq176_HTML.gif is a continuous and bounded operator.

3. Existence and Multiplicity of Solutions

Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ15_HTML.gif
(3.1)

The critical points of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq177_HTML.gif , that is,

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ16_HTML.gif
(3.2)

for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq178_HTML.gif , are weak solutions of problem (1.1). So we need only to consider the existence of nontrivial critical points of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq179_HTML.gif .

Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq180_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq181_HTML.gif the generic positive constants. Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq182_HTML.gif the Lebesgue measure of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq183_HTML.gif .

To study the existence of solutions for problem (1.1) in the first case, we additionally impose the following conditions.

(A-1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq184_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq185_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq186_HTML.gif .

(B-1) There exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq187_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq188_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq189_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq190_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq191_HTML.gif .

(C-1) there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq192_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq193_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq194_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq195_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq196_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq197_HTML.gif .

(D-1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq198_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq199_HTML.gif .

Theorem 3.1.

Under assumptions (F) and (A-1)–(C-1), problem (1.1) admits a nontrivial solution.

Proof.

First we show that any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq200_HTML.gif sequence is bounded. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq201_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq202_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq203_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq204_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq205_HTML.gif . By (A-1) and (B-1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq206_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq207_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq208_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq209_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq210_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq211_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq212_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq214_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq215_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq216_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ17_HTML.gif
(3.3)

By (B-1), we get

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ18_HTML.gif
(3.4)

By (F), we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq217_HTML.gif , so there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq218_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq219_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq220_HTML.gif . Note http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq221_HTML.gif , so we have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ19_HTML.gif
(3.5)

By Young's inequality, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq222_HTML.gif , we get

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ20_HTML.gif
(3.6)

Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq223_HTML.gif sufficiently small so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq224_HTML.gif .

Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq225_HTML.gif , by Young's inequality again and for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq226_HTML.gif , we get

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ21_HTML.gif
(3.7)

Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq227_HTML.gif sufficiently small so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq228_HTML.gif .

From the above remark, we have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ22_HTML.gif
(3.8)

As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq229_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq230_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq231_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq232_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq233_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq234_HTML.gif , by Theorem 2.6 we have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ23_HTML.gif
(3.9)

when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq235_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq236_HTML.gif is sufficiently large. Then it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq237_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq238_HTML.gif . Next we show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq239_HTML.gif possesses a convergent subsequence (still denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq240_HTML.gif ).

Note that

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ24_HTML.gif
(3.10)
Because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq241_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq242_HTML.gif , there exists a subsequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq243_HTML.gif (still denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq244_HTML.gif ), such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq245_HTML.gif weakly in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq246_HTML.gif . By Theorem 2.11, there are compact embeddings http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq247_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq248_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq249_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq251_HTML.gif . So we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ25_HTML.gif
(3.11)

Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq252_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq253_HTML.gif .

By (F), we have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ26_HTML.gif
(3.12)
and similarly for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq254_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ27_HTML.gif
(3.13)
Since
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ28_HTML.gif
(3.14)
and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq255_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq256_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq257_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq258_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq259_HTML.gif . Similarly,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ29_HTML.gif
(3.15)

Because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq260_HTML.gif is bounded, we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq261_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq262_HTML.gif . From the above remark, we conclude http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq263_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq264_HTML.gif .

Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq265_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq266_HTML.gif . Then we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq267_HTML.gif . As in the proof of Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq268_HTML.gif in [6, 7], we divide http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq269_HTML.gif into the following two parts:

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ30_HTML.gif
(3.16)

On http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq270_HTML.gif , we have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ31_HTML.gif
(3.17)

Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq271_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq272_HTML.gif .

On http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq273_HTML.gif , we have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ32_HTML.gif
(3.18)

so http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq274_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq275_HTML.gif .

Thus we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq276_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq277_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq278_HTML.gif .

From (F) and (B-1) we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq279_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq280_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq281_HTML.gif . So we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq282_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq283_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq284_HTML.gif . for  all   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq285_HTML.gif , take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq286_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq287_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq288_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq289_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq290_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq291_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq292_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq293_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq294_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq295_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq296_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq297_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq298_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq299_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq300_HTML.gif . Then we have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ33_HTML.gif
(3.19)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq301_HTML.gif . So if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq302_HTML.gif is sufficiently large, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq303_HTML.gif .

From (F) and (C-1), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq304_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq305_HTML.gif . So we get

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ34_HTML.gif
(3.20)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq306_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq307_HTML.gif . By Theorem 2.11, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq308_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq309_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq310_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq311_HTML.gif is sufficiently small, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq312_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq313_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq314_HTML.gif . For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq315_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq316_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq317_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq318_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq319_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq320_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq321_HTML.gif . Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq322_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq323_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq324_HTML.gif is an open covering of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq325_HTML.gif . As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq326_HTML.gif is compact, we can pick a finite subcovering http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq327_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq328_HTML.gif from the covering http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq329_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq330_HTML.gif we define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq331_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq332_HTML.gif . We can use all the hyperplanes, for each of which there exists at least one hypersurface of some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq333_HTML.gif lying on it, to divide http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq334_HTML.gif into finite open hypercubes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq335_HTML.gif which mutually have no common points. It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq336_HTML.gif and for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq337_HTML.gif there exists at least one http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq338_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq339_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq340_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq341_HTML.gif and we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ35_HTML.gif
(3.21)
If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq342_HTML.gif is sufficiently small such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ36_HTML.gif
(3.22)

we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq343_HTML.gif .

The mountain pass theorem guarantees that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq344_HTML.gif has a nontrivial critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq345_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq346_HTML.gif is a separable and reflexive Banach space, there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq347_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq348_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ37_HTML.gif
(3.23)

For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq349_HTML.gif , denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq350_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq351_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq352_HTML.gif .

Theorem 3.2.

Under assumptions (F), (A-1)–(D-1), problem (1.1) admits a sequence of solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq353_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq354_HTML.gif .

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq355_HTML.gif . We first show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq356_HTML.gif is weakly strongly continuous. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq357_HTML.gif weakly in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq358_HTML.gif . By the compact embedding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq359_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq360_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq361_HTML.gif a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq362_HTML.gif . By the inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq363_HTML.gif and the Vitali Theorem, we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq364_HTML.gif .

Note that

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ38_HTML.gif
(3.24)
When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq365_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ39_HTML.gif
(3.25)
When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq366_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq367_HTML.gif is bounded. So we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ40_HTML.gif
(3.26)
By the compact embedding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq368_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq369_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq370_HTML.gif . So by Theorem 2.12 we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq371_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq372_HTML.gif . Hence we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq373_HTML.gif is weakly strongly continuous. By Proposition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq374_HTML.gif in [8], http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq375_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq376_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq377_HTML.gif . For all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq378_HTML.gif , there exists a positive integer http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq379_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq380_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq381_HTML.gif . Assume http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq382_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq383_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq384_HTML.gif in the following way:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ41_HTML.gif
(3.27)
Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq385_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq386_HTML.gif . Hence for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq387_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq388_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ42_HTML.gif
(3.28)
So
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ43_HTML.gif
(3.29)

Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq389_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq390_HTML.gif . Since the dimension of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq391_HTML.gif is finite, any two norms on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq392_HTML.gif are equivalent, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq393_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq394_HTML.gif , it is immediate that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq395_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq396_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq397_HTML.gif . As in the proof of Theorem 3.1 we can find hypercubes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq398_HTML.gif which mutually have no common points such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq399_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq400_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq401_HTML.gif . Then we need only to consider the case: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq402_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq403_HTML.gif . We have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ44_HTML.gif
(3.30)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq404_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq405_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq406_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq407_HTML.gif . Denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq408_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq409_HTML.gif be sufficiently large such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq410_HTML.gif . There at least exists one http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq411_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq412_HTML.gif . We have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ45_HTML.gif
(3.31)

and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq413_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq414_HTML.gif . Hence we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq415_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq416_HTML.gif . Thus for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq417_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq418_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq419_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq420_HTML.gif . From Theorem 3.1   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq421_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq422_HTML.gif condition. In view of (D-1), by Fountain Theorem (see [16]), we conclude the result.

In the second case, we additionally impose the following condition:

(A-2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq423_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq424_HTML.gif .

Theorem 3.3.

Under assumptions (F), (A-2), (B-1), and (C-1) there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq425_HTML.gif such that when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq426_HTML.gif , problem (1.1) admits a nontrivial solution.

Proof.

It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq427_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq428_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq429_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq430_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq431_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq432_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq433_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq434_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq435_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq436_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq437_HTML.gif be as defined in Theorem 3.1. By (C-1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq438_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq439_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq440_HTML.gif . Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq441_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq442_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ46_HTML.gif
(3.32)

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq443_HTML.gif is sufficiently small, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq444_HTML.gif .

From (F) and (C-1), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq445_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq446_HTML.gif . By Theorems 2.8 and 2.11, there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq447_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq448_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq449_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq450_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq451_HTML.gif is sufficiently small, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq452_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq453_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq454_HTML.gif . As in the proof of Theorem 3.1 we can find hypercubes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq455_HTML.gif which mutually have no common points such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq456_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq457_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq458_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq459_HTML.gif . Then

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ47_HTML.gif
(3.33)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq460_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq461_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq462_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq463_HTML.gif . Fix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq464_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq465_HTML.gif . Then we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ48_HTML.gif
(3.34)

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq466_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq467_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq468_HTML.gif . As in the proof of Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq469_HTML.gif in [17], denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq470_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq471_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq472_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq473_HTML.gif . Applying Ekeland's variational principle to the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq474_HTML.gif , we find http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq475_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq476_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq477_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq478_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq479_HTML.gif . Thus we get a sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq480_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq481_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq482_HTML.gif . It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq483_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq484_HTML.gif . As in the proof of Theorem 3.1, we get a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq485_HTML.gif , still denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq486_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq487_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq488_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq489_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq490_HTML.gif .

Theorem 3.4.

Under assumptions (F), (A-2), and (B-1)–(D-1), problem (1.1) has a sequence of solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq491_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq492_HTML.gif .

Proof.

First we show that any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq493_HTML.gif sequence is bounded. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq494_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq495_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq496_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq497_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq498_HTML.gif . By (B-1), there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq499_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq500_HTML.gif . From (F), (A-2), and (B-1)–(D-1), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ49_HTML.gif
(3.35)
Thus we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ50_HTML.gif
(3.36)

By Young's inequality, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq501_HTML.gif , we get

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ51_HTML.gif
(3.37)
Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq502_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq503_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq504_HTML.gif sufficiently small so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq505_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq506_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq507_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ52_HTML.gif
(3.38)
Therefore by Theorems 2.6 and 2.9, we get that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq508_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq509_HTML.gif . Then as in the proof of Theorem 3.1 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq510_HTML.gif possesses a convergent subsequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq511_HTML.gif (still denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq512_HTML.gif ). By Theorem 3.2, we can also have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ53_HTML.gif
(3.39)

As in the proof of Theorem 3.1 we can find hypercubes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq513_HTML.gif which mutually have no common points such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq514_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq515_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq516_HTML.gif . Since the dimension of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq517_HTML.gif is finite, any two norms on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq518_HTML.gif are equivalent. Then we need only to consider the cases http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq519_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq520_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq521_HTML.gif . We have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ54_HTML.gif
(3.40)

Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq522_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq523_HTML.gif . As in the proof of Theorem 3.2, we complete the proof.

In the third case, we additionally impose the following condition:

(A-3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq524_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq525_HTML.gif ,

(B-3) there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq526_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq527_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq528_HTML.gif .

Theorem 3.5.

Under assumptions (F), (A-3), and (B-3), problem (1.1) admits a nontrivial solution.

Proof.

By Young's inequality, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq529_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq530_HTML.gif . By (F), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq531_HTML.gif . Thus
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ55_HTML.gif
(3.41)
Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq532_HTML.gif sufficiently small so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq533_HTML.gif . From Theorem 2.11, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq534_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq535_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq536_HTML.gif is bounded. Then we need only to consider the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq537_HTML.gif . As in the proof of Theorem 3.1 we can find hypercubes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq538_HTML.gif which mutually have no common points such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq539_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq540_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq541_HTML.gif . Then we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ56_HTML.gif
(3.42)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq542_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq543_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq544_HTML.gif . As in the proof of Theorem 3.2, we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq545_HTML.gif is coercive, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq546_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq547_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq548_HTML.gif has a critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq549_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq550_HTML.gif and further http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq551_HTML.gif is a weak solution of (1.1).

Next we show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq552_HTML.gif is nontrivial. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq553_HTML.gif be the same as that in Theorem 3.3. By (B-3), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq554_HTML.gif . Then

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ57_HTML.gif
(3.43)

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq555_HTML.gif is sufficiently small, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq556_HTML.gif .

In the fourth case, we additionally impose the following condition:

(A-4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq557_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq558_HTML.gif .

Theorem 3.6.

Under assumptions (F), (A-4), and (D-1), problem (1.1) admits a sequence of solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq559_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq560_HTML.gif .

Proof.

First we show that any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq561_HTML.gif sequence is bounded. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq562_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq563_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq564_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq565_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq566_HTML.gif . Denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq567_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq568_HTML.gif . We have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq569_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq570_HTML.gif .

We can get

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ58_HTML.gif
(3.44)
By Young's inequality, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq571_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ59_HTML.gif
(3.45)
Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq572_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq573_HTML.gif sufficiently small so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq574_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq575_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ60_HTML.gif
(3.46)
As in the proof of Theorem 3.5, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq576_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq577_HTML.gif . Thus, we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq578_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq579_HTML.gif . Then as in the proof of Theorem 3.1 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq580_HTML.gif possesses a convergent subsequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq581_HTML.gif (still denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq582_HTML.gif ). By Theorem 3.2, we can also get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ61_HTML.gif
(3.47)

As in the proof of Theorem 3.1 we can find hypercubes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq583_HTML.gif which mutually have no common points such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq584_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq585_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq586_HTML.gif . Since the dimension of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq587_HTML.gif is finite, any two norms on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq588_HTML.gif are equivalent. Then we need only to consider the cases http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq589_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq590_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq591_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq592_HTML.gif . We have

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ62_HTML.gif
(3.48)

Hence we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq593_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq594_HTML.gif . As in the proof of Theorem 3.2, we complete the proof.

Declarations

Acknowledgments

This work is supported by the Science Research Foundation in Harbin Institute of Technology (HITC200702), The Natural Science Foundation of Heilongjiang Province (A2007-04), and the Program of Excellent Team in Harbin Institute of Technology.

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology

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Copyright

© F. Yongqiang and Y. Mei. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.