Existence of a Positive Solution for a Nonlinear Elliptic Equation with Saddle-like Potential and Nonlinearity with Exponential Critical Growth in $${\mathbb{R}^{2}}$$ R 2

Abstract

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