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