Electron. J. Diff. Equ., Vol. 2010(2010), No. 44, pp. 1-9.

Existence and multiplicity of solutions for a differential inclusion problem involving the p(x)-Laplacian

Guowei Dai

Abstract:
In this article we consider the differential inclusion
$$\displaylines{ -\hbox{div}(|\nabla u|^{p(x)-2}\nabla u)\in \partial F(x,u) \quad\hbox{in }\Omega,\cr u=0 \quad \hbox{on }\partial \Omega }$$
which involves the $p(x)$-Laplacian. By applying the nonsmooth Mountain Pass Theorem, we obtain at least one nontrivial solution; and by applying the symmetric Mountain Pass Theorem, we obtain k-pairs of nontrivial solutions in $W_{0}^{1,p(x)}(\Omega)$.

Submitted December 31, 2009. Published March 26, 2010.
Math Subject Classifications: 35J20, 35J70, 35R70.
Key Words: p(x)-Laplacian; nonsmooth mountain pass theorem; differential inclusion.

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Guowei Dai
Department of Mathematics, Northwest Normal University
Lanzhou, 730070, China
email: daigw06@lzu.cn

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