Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 25, pp. 1-12. Title: Existence of solutions for discontinuous p(x)-Laplacian problems with critical exponents Authors: Xudong Shang (Nanjing Normal Univ., Jiangsu, China) Zhigang Wang (Nanjing Normal Univ., Jiangsu, China) Abstract: In this article, we study the existence of solutions to the problem $$\displaylines{ -\hbox{div}(|\nabla u|^{p(x)-2}\nabla u) =\lambda |u|^{p^{*}(x)-2}u + f(u)\quad x \in \Omega ,\cr u = 0 \quad x \in \partial\Omega, }$$ where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^{N}$, $p(x)$ is a continuous function with $1