Existence of standing waves solution for a Nonlinear Schr\"odinger equations in $\mathbb{R}^{N}$

Abstract

In this paper, we investigate the existence of positive solution for the following class of elliptic equation $$ - \epsilon^{2}\Delta u +V(x)u= f(u) \,\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon >0$ is a positive parameter, $f$ has a subcritical growth and $V$ is a positive potential verifying some conditions.