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

TL;DR

This paper proves the existence of positive solutions for a nonlinear elliptic equation in ^{2} with a saddle-like potential and exponential critical growth, using variational methods.

Contribution

It establishes the existence of solutions under saddle-like potential and exponential growth conditions, advancing understanding of elliptic equations in ^{2}.

Findings

Existence of positive solutions proven using variational methods.

Handles exponential critical growth in ^{2}.

Addresses saddle-like potential scenarios.

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.