Existence of solutions for the \(p(x)\)-Laplacian problem with the critical Sobolev-Hardy exponent (Q448844)

Summary: This paper deals with the \(p(x)\)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in \(W^{1,p(x)}_0 (\Omega)\) space is established, then by applying it we obtain the existence of solutions for the following \(p(x)\)-Laplacian problem: \(-\text{div}(|\nabla u|^{p(x)-2} \nabla u) + |u|^{p(x)-2}u = (h(x)|u|^{p^\ast_s(x)-2} u/|x|^{s(x)}) + f(x, u), x \in \Omega, u = 0, x \in \partial\Omega\), where \(\Omega \subset \mathbb R^N\) is a bounded domain, \(0 \in \Omega, 1 < p^- \leq p(x) \leq p^+ < N\), and \(f(x, u)\) satisfies \(p(x)\)-growth conditions.