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

TL;DR

This paper proves the existence of solutions for a class of multivalued elliptic equations with exponential critical growth in two-dimensional space using variational methods, especially when the parameter epsilon is small.

Contribution

It introduces a novel application of variational methods to multivalued elliptic problems with exponential critical growth and discontinuities in \\mathbb{R}^2.

Findings

Existence of two solutions for small epsilon

Application of variational methods to multivalued problems

Handling exponential critical growth and discontinuities

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.