Existence of positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in $\mathbb{R}^{2}$

Abstract: In this paper, we use variational methods to prove the existence of positive solution for the following class of elliptic equation $$ -\epsilon^{2}\Delta{u}+V(z)u=f(u) \,\,\, \mbox{in} \,\,\, \mathbb{R}^{2}, $$ where $\epsilon >0$ is a positive parameter, $V$ is a saddle-like potential and $f$ has an exponential critical growth.