Existence of Solutions for thep(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent

Yu Mei,Li Wang,Fu Yongqiang

doi:10.1155/2012/894925

Abstract

This paper deals with thep(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness inW01,p(x)(Ω)space is established, then by applying it we obtain the existence of solutions for the followingp(x)-Laplacian problem:-div (|∇u|p(x)-2∇u)+|u|p(x)-2u=(h(x)|u|ps*(x)-2u/|x|s(x))+f(x,u), x∈Ω, u=0, x∈∂Ω,whereΩ⊂ℝNis a bounded domain,0∈Ω,1<p-≤p(x)≤p+<N, andf(x,u)satisfiesp(x)-growth conditions.

Highlights

In this paper we are concerned with the following p x -Laplacian problem:
When p x ≡ p is a constant function, the p-Laplacian problem related to Sobolev-Hardy inequality had been studied by many authors, either is the bounded domain or in the whole space RN, see, for example, 1–4
This paper is organized as follows: in Section 2 we deal with some preliminary materials and technical results; in Section 3 we give the proof of a principle of concentration compactness; in Section 4 we study the problem of p x -Laplacian equation with the critical Sobolev-Hardy exponent

Summary

Introduction

In this paper we are concerned with the following p x -Laplacian problem:. 0, x ∈ ∂Ω, 1.1 where 0 ∈ Ω ⊂ RN is a bounded domain, p x is Lipschitz continuous, radially symmetric on. In this paper we are concerned with the following p x -Laplacian problem:. We explain some notations employed in this paper: Let P Ω be the set of all Lebesgue measurable functions p : Ω → 1, ∞. When p x ≡ p is a constant function, the p-Laplacian problem related to Sobolev-. Hardy inequality had been studied by many authors, either is the bounded domain or in the whole space RN, see, for example, 1–4. This paper is organized as follows: in Section 2 we deal with some preliminary materials and technical results; in Section 3 we give the proof of a principle of concentration compactness; in Section 4 we study the problem of p x -Laplacian equation with the critical Sobolev-Hardy exponent

Preliminaries

The Principle of Concentration Compactness

Existence of Solutions

10 Note that