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 $$ -\Delta u+V(x)u-\epsilon h(x)\in \partial_t F(x,u) \quad \text{in} \quad \mathbb{R}^2, \eqno{(P)} $$ where $\epsilon>0$, $V$ is a continuous function verifying some conditions, $h \in (H^{1}(\mathbb{R}^{2}))^{*}$ 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.