Existence of solutions for p(x)-Laplacian Dirichlet problem

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Abstract

This paper presents several sufficient conditions for the existence of solutions for the Dirichlet problem of p(x)-Laplacian−div(|u|p(x)−2u)=f(x,u),x∈Ω,u=0,x∈∂Ω.Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained. The discussion is based on the theory of the spaces Lp(x)(Ω) and W01,p(x)(Ω).

Introduction

The study of variational problems with nonstandard growth conditions is an interesting topic in recent years. p(x)-growth conditions can be regarded as an important case of nonstandard (p,q)-growth conditions. Many results have been obtained on this kind of problems, for example [2], [3], [4], [6], [7], [9], [17], [18]. Under (p,q)-growth conditions Marcellini [9] proved the existence and regularity of weak solutions of elliptic equations of divergence form with differentiable coefficients in the nondegenerate case. Under p(x)-growth conditions, Fan and Zhao [4], [6], [7] proved that the weak solutions of elliptic equations are Hölder continuous and its gradients have high integrability.

In this paper, we consider the existence of weak solutions of the problem(P)−△p(x)u≔−div(|u|p(x)−2u)=f(x,u),x∈Ω,u=0,x∈∂Ω,where Ω⊂RN is a bounded domain, 1<p(x), and p(x)∈C(Ω̄). Our aim is to give several existence results of weak solutions for problem (P). These results are extensions of that of p-Laplacian problems.

This paper is divided into four sections. In the second section, we introduce some basic properties of the generalized Lebesgue–Sobolev spaces W01,p(x)(Ω) which can be regarded as a special class of generalized Orlicz–Sobolev spaces. In the third section, several important properties of p(x)-Laplace operator are presented. Finally, in the fourth section, we give some existence results of weak solutions of problem (P).

Section snippets

The spaces W01,p(x)(Ω)

In order to discuss problem (P), we need some theories on spaces W01,p(x)(Ω) which we call generalized Lebesgue–Sobolev spaces. Firstly we state some basic properties of spaces W01,p(x)(Ω) which will be used later (for details, see [5], [10], [16], [15]). WriteC+(Ω̄)={h|h∈C(Ω̄),h(x)>1foranyx∈Ω̄},h+=maxΩ̄h(x),h=minΩ̄h(x)foranyh∈C(Ω̄),Lp(x)(Ω)=u|uisameasurablereal-valuedfunction,Ω|u(x)|p(x)dx<∞.We can introduce the norm on Lp(x)(Ω) by|u|p(x)=infλ>0Ωu(x)λp(x)dx⩽1and (Lp(x)(Ω), |·|p(x)) becomes

Properties of p(x)-Laplace operator

In this section, we discuss the p(x)-Laplace operator −△p(x)u≔−div(|∇u|p(x)−2u). Consider the following functional:J(u)=Ω1p(x)|u|p(x)dx,u∈X≔W01,p(x)(Ω).

We know that (see [1]), JC1(X,R), and the p(x)-Laplace operator is the derivative operator of J in the weak sense. We denote L=J′:X→X, then(L(u),v)=Ω|u|p(x)−2uvdx∀v,u∈X.

Theorem 3.1

(i) L:X→X is a continuous, bounded and strictly monotone operator;

(ii) L is a mapping of type (S+), i.e. if un⇀u in X and limn→∞(L(un)−L(u),un−u)⩽0, then unu in X;

Existence of solutions

In this section we will discuss the existence of weak solutions of (P).

Definition 4.1

We call that u∈W01,p(x)(Ω) is a weak solution of (P), ifΩ|u|p(x)−2vdx=Ωf(x,u)vdx∀v∈X≔W01,p(x)(Ω).

If f is independent of u, we have

Theorem 4.2

If f(x,u)=f(x), f∈Lα(x)(Ω), where α∈C+(Ω̄), satisfies 1/α(x)+1/p(x)<1, then (P) has a unique weak solution.

Proof

According to Proposition 2.5(ii), (f,v)≔Ωf(x)vdx (for any vX) defines a continuous linear functional on X. Since L is a homeomorphism, (P) has a unique solution.

Hereafter, f(x,t) is

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Research supported by the National Science Foundation of China (19971036) and the Natural Science Foundation of Gansu Province (ZS991-A25-005-Z).

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