Multivalued Elliptic Equation with exponential critical growth in ${\mathbb{R}}^2$

Abstract

In this work we study the existence of nontrivial solution for the following class of multivalued elliptic problems $$ -Δu+V(x)u-εh(x)\in \partial_t F(x,u) \quad \text{in} \quad \mathbb{R}^2, \eqno{(P)} $$ where $\epsilon >0$, $V$ is a continuous function verifying some conditions, $h\in (H^1({\mathbb{R}}^2){)}^{\ast }$ and ${\partial }_{t}F(x,u)$ is a generalized gradient of $F(x,t)$ with respect to $t$ and $F(x,t)={\int }_0^{t}f(x,s)ds$. Assuming that $f$ has an exponential critical growth and a discontinuity point, we have applied Variational Methods for locally Lipschitz functional to get two solutions for $(P)$ when $\epsilon$ is small enough.