Electron. J. Diff. Equ., Vol. 2012 (2012), No. 25, pp. 1-12.

Existence of solutions for discontinuous p(x)-Laplacian problems with critical exponents

Xudong Shang, Zhigang Wang

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 $1<p(x)<N$ and $p^{*}(x) = \frac{Np(x)}{N-p(x)}$. Applying nonsmooth critical point theory for locally Lipschitz functionals, we show that there is at least one nontrivial solution when $\lambda$ less than a certain number, and $f$ maybe discontinuous.

Submitted November 7, 2011. Published February 7, 2012.
Math Subject Classifications: 35J92, 35J70, 35R70
Key Words: p(x)-Laplacian problem; critical Sobolev exponents; discontinuous nonlinearities.

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Xudong Shang
School of Mathematics
Nanjing Normal University Taizhou College
225300, Jiangsu, China
email: xudong-shang@163.com
Zhigang Wang
School of Mathematics
Nanjing Normal University Taizhou College
225300, Jiangsu, China
email: wzg19.scut@163.com

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