On a Nonlocal Multivalued Problem in an Orlicz-Sobolev Space via Krasnoselskii's Genus

This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem involving N-functions and theory of locally Lispchitz continuous functionals. More precisely, in this paper, we study a result of multiplicity to the following multivalued elliptic problem: $$\left \{ \begin{array}{l} -M\left(\displaystyle\int_\Omega \Phi(\mid\nabla u\mid)dx\right) div\big(\phi(\mid\nabla u\mid)\nabla u\big) -\phi(|u|)u\in \partial F(u) \ \mbox{in}\ \Omega,\\[6mm] u\in W_0^1L_\Phi(\Omega), \end{array} \right.$$ where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain, $N\geq 2$, $M$ is continuous function, $\Phi$ is an N-function with $\Phi(t)=\displaystyle\int^{|t|}_{0}\phi(s)s \ ds$ and $\partial F(t)$ is a generalized gradient of $F(t)$. We use genus theory to obtain the main result