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: $-\text{div}\hspace{0.17em}(|\nabla u{|}^{p(x)-2}\nabla u)+|u{|}^{p(x)-2}u=(h(x)|u{|}^{p_s^{*}(x)-2}u/|x{|}^{s(x)})+f(x,u),\hspace{0.17em}\hspace{0.17em}x\in \mathrm{\Omega },\hspace{0.17em}\hspace{0.17em}u=0,\hspace{0.17em}\hspace{0.17em}x\in \partial \mathrm{\Omega },$ where $\mathrm{\Omega }\subset {ℝ}^N$ is a bounded domain, $0\in \mathrm{\Omega }$, , and $f(x,u)$ satisfies $p(x)$-growth conditions.