Strongly nonlinear multivalued elliptic equations on a bounded domain

In this work we study the existence of nontrivial solution for the following class of multivalued quasilinear problems <Equation ID="Equ51"> <EquationSource Format="TEX">$$\begin{aligned} \displaystyle -\text{ div } ( \phi (|\nabla u|) \nabla u) - b(u)u \in \lambda \partial F(x,u)\;\text{ in }\;\Omega , \quad u=0\; \text{ on }\;\partial \Omega \end{aligned}$$</EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\Omega \subset \mathbb{R }^N$$</EquationSource> </InlineEquation> is a bounded domain, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$N\ge 2$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\partial F(x,u)$$</EquationSource> </InlineEquation> is a generalized gradient of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$F(x,t)$$</EquationSource> </InlineEquation> with respect to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$t$$</EquationSource> </InlineEquation>. The main tools utilized are Variational Methods for Locally Lipschitz Functional and a Concentration Compactness Theorem for Orlicz space. Copyright Springer Science+Business Media New York 2014