Open Access

Existence of Solutions for the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq2_HTML.gif -Laplacian problem with singular term: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq6_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq7_HTML.gif is a bounded domain, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq10_HTML.gif spaces and https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq15_HTML.gif -Laplacian problem:

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

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

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

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

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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq38_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq39_HTML.gif . In fact, take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq41_HTML.gif , and https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq43_HTML.gif ;

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

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

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

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq47_HTML.gif , the solution of the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq49_HTML.gif and variable exponent Sobolev space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq50_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq52_HTML.gif and

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq53_HTML.gif is the class of functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq54_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq55_HTML.gif . https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq57_HTML.gif we define the conjugate function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq58_HTML.gif as

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

Theorem 2.1.

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

holds for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq61_HTML.gif with the constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq62_HTML.gif depending on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq63_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq65_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq66_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq67_HTML.gif . The space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq68_HTML.gif is reflexive if and only if
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq69_HTML.gif satisfies (2.4). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq71_HTML.gif , then necessary and sufficient condition for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq72_HTML.gif is that for almost all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq73_HTML.gif one has https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq75_HTML.gif satisfies (2.4). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq76_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq77_HTML.gif , then

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

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

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

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

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

We assume that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq89_HTML.gif , we set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq91_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq93_HTML.gif is the class of functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq94_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq95_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq96_HTML.gif for every multi-index https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq97_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq98_HTML.gif , endowed with the norm

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

By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq99_HTML.gif we denote the subspace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq100_HTML.gif which is the closure of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq102_HTML.gif :

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq103_HTML.gif .

Theorem 2.5.

The spaces https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq105_HTML.gif are separable reflexive Banach spaces, if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq107_HTML.gif satisfies (2.4). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq108_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq109_HTML.gif , then

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

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

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

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

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

We denote the dual space of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq120_HTML.gif by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq122_HTML.gif . Then for every https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq124_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ9_HTML.gif
(2.7)
The norm of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq125_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ10_HTML.gif
(2.8)

Theorem 2.8.

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

Theorem 2.9.

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

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq137_HTML.gif is a constant depending on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq139_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq140_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq141_HTML.gif . Define

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

with the norm

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

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

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

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

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

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

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

(6) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq158_HTML.gif if and only if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq160_HTML.gif possesses the cone property and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq161_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq162_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq163_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq164_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq166_HTML.gif then there is a compact embedding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq167_HTML.gif .

Theorem 2.12.

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

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq171_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq172_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq174_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq175_HTML.gif defined by https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ15_HTML.gif
(3.1)

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

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

for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq179_HTML.gif .

Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq180_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq181_HTML.gif the generic positive constants. Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq182_HTML.gif the Lebesgue measure of https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq184_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq185_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq186_HTML.gif .

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

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

(D-1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq198_HTML.gif , for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq200_HTML.gif sequence is bounded. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq201_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq202_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq203_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq204_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq205_HTML.gif . By (A-1) and (B-1), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq206_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq207_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq208_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq209_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq210_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq211_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq212_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq213_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq214_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq216_HTML.gif . Then
https://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

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq222_HTML.gif , we get

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

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

Note that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq226_HTML.gif , we get

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

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

From the above remark, we have

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

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

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

when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq235_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq237_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq238_HTML.gif . Next we show that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq240_HTML.gif ).

Note that

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

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

By (F), we have

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

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

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

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

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

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

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

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

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

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

Thus we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq276_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq277_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq279_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq280_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq281_HTML.gif . So we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq282_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq283_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq284_HTML.gif . for  all   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq285_HTML.gif , take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq286_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq287_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq288_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq289_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq290_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq291_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq292_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq293_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq294_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq295_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq296_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq297_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq298_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq299_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq300_HTML.gif . Then we have

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

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq301_HTML.gif . So if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq302_HTML.gif is sufficiently large, we obtain https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq304_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq305_HTML.gif . So we get

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ34_HTML.gif
(3.20)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq306_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq307_HTML.gif . By Theorem 2.11, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq308_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq309_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq310_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq311_HTML.gif is sufficiently small, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq312_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq313_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq314_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq315_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq316_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq317_HTML.gif , we can find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq318_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq319_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq320_HTML.gif whenever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq321_HTML.gif . Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq322_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq323_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq324_HTML.gif is an open covering of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq325_HTML.gif . As https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq327_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq328_HTML.gif from the covering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq329_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq330_HTML.gif we define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq331_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq333_HTML.gif lying on it, to divide https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq334_HTML.gif into finite open hypercubes https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq336_HTML.gif and for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq337_HTML.gif there exists at least one https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq338_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq339_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq340_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq341_HTML.gif and we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ35_HTML.gif
(3.21)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq342_HTML.gif is sufficiently small such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ36_HTML.gif
(3.22)

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

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

Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq347_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq348_HTML.gif such that

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

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq349_HTML.gif , denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq350_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq351_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq353_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq354_HTML.gif .

Proof.

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

Note that

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

Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq389_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq390_HTML.gif . Since the dimension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq391_HTML.gif is finite, any two norms on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq392_HTML.gif are equivalent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq393_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq394_HTML.gif , it is immediate that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq395_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq396_HTML.gif , then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq399_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq400_HTML.gif , where https://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: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq402_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq403_HTML.gif . We have

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

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq413_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq414_HTML.gif . Hence we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq415_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq416_HTML.gif . Thus for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq417_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq418_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq419_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq420_HTML.gif . From Theorem 3.1   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq421_HTML.gif satisfies https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq423_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq425_HTML.gif such that when https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq427_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq428_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq429_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq430_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq431_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq432_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq433_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq434_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq435_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq436_HTML.gif . Let https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq438_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq439_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq440_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq441_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq442_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ46_HTML.gif
(3.32)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq443_HTML.gif is sufficiently small, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq445_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq447_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq448_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq449_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq450_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq451_HTML.gif is sufficiently small, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq452_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq453_HTML.gif , and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq456_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq457_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq458_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq459_HTML.gif . Then

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq466_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq467_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq468_HTML.gif . As in the proof of Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq469_HTML.gif in [17], denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq470_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq471_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq472_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq474_HTML.gif , we find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq475_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq476_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq477_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq478_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq479_HTML.gif . Thus we get a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq480_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq481_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq482_HTML.gif . It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq483_HTML.gif is bounded in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq485_HTML.gif , still denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq486_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq487_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq488_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq489_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq491_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq492_HTML.gif .

Proof.

First we show that any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq493_HTML.gif sequence is bounded. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq494_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq495_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq496_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq497_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq498_HTML.gif . By (B-1), there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq499_HTML.gif such that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ49_HTML.gif
(3.35)
Thus we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq501_HTML.gif , we get

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ51_HTML.gif
(3.37)
Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq502_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq503_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq504_HTML.gif sufficiently small so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq505_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq506_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq507_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq508_HTML.gif is bounded in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq510_HTML.gif possesses a convergent subsequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq511_HTML.gif (still denoted by https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq514_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq515_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq516_HTML.gif . Since the dimension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq517_HTML.gif is finite, any two norms on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq519_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq520_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq521_HTML.gif . We have

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

Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq522_HTML.gif as https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq524_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq525_HTML.gif ,

(B-3) there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq526_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq527_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq529_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq530_HTML.gif . By (F), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq531_HTML.gif . Thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ55_HTML.gif
(3.41)
Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq532_HTML.gif sufficiently small so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq533_HTML.gif . From Theorem 2.11, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq534_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq535_HTML.gif , https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq539_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq540_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq541_HTML.gif . Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ56_HTML.gif
(3.42)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq542_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq543_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq545_HTML.gif is coercive, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq546_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq547_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq548_HTML.gif has a critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq549_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq550_HTML.gif and further https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq552_HTML.gif is nontrivial. Let https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq554_HTML.gif . Then

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

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq555_HTML.gif is sufficiently small, https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq557_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq559_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq560_HTML.gif .

Proof.

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

We can get

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq571_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_Equ59_HTML.gif
(3.45)
Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq572_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq573_HTML.gif sufficiently small so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq574_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq575_HTML.gif . Then
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq576_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq577_HTML.gif . Thus, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq578_HTML.gif is bounded in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq580_HTML.gif possesses a convergent subsequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq581_HTML.gif (still denoted by https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq584_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq585_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq586_HTML.gif . Since the dimension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq587_HTML.gif is finite, any two norms on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq589_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq590_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq591_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq592_HTML.gif . We have

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

Hence we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F584843/MediaObjects/13661_2009_Article_941_IEq593_HTML.gif as https://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.