## Abstract and Applied Analysis

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

#### Abstract

This paper deals with the $p(x)$-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in ${W}_{0}^{1,p(x)}(\mathrm{\Omega })$ space is established, then by applying it we obtain the existence of solutions for the following $p(x)$-Laplacian problem: $-\text{div} (|\nabla u{|}^{p(x)-2}\nabla u)+|u{|}^{p(x)-2}u=(h(x)|u{|}^{{p}_{s}^{\ast}(x)-2}u/|x{|}^{s(x)})+f(x,u), x\in \mathrm{\Omega }, u=0, x\in \partial \mathrm{\Omega },$ where $\mathrm{\Omega }\subset {\Bbb R}^{N}$ is a bounded domain, $0\in \mathrm{\Omega }$, $1<{p}^{-}\le p(x)\le {p}^{+}, and $f(x,u)$ satisfies $p(x)$-growth conditions.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 894925, 17 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495827

Digital Object Identifier
doi:10.1155/2012/894925

Mathematical Reviews number (MathSciNet)
MR2965446

Zentralblatt MATH identifier
1250.35107

#### Citation

Mei, Yu; Yongqiang, Fu; Wang, Li. Existence of Solutions for the $p(x)$ -Laplacian Problem with the Critical Sobolev-Hardy Exponent. Abstr. Appl. Anal. 2012 (2012), Article ID 894925, 17 pages. doi:10.1155/2012/894925. https://projecteuclid.org/euclid.aaa/1355495827

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