On Variational Multivalued Elliptic Equations on a Bounded Domain in the Presence of Critical Growth

Abstract

We develop arguments on the critical point theory for locally Lipschitz functionals on Orlicz-Sobolev spaces, along with convexity and compactness techniques to investigate existence of solution of the multivalued equation $\displaystyle - \Delta_{\Phi} u \in \partial j(.,u) + \lambda h \mbox{in} \Omega$, where $\Omega \subset {\bf R}^{N}$ is a bounded smooth domain, $\Phi : {\r} \longrightarrow [0,\infty)$ is a suitable N-function, $\Delta_{\Phi}$ is the corresponding $\Phi$-Laplacian, $\lambda > 0$ is a parameter, $h:\Omega\rightarrow{\r}$ is integrable and $\partial j(., u)$ is the subdifferential of a function $j$ associated with critical growth.