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

TL;DR

This paper proves the existence of positive standing wave solutions for a class of nonlinear Schrödinger equations in multi-dimensional space, considering variable potentials and subcritical nonlinearities.

Contribution

It establishes new existence results for positive solutions under specific conditions on the potential and nonlinearity, extending previous work to broader settings.

Findings

Positive solutions exist for small epsilon values.

Solutions are localized and positive.

Conditions on V and f ensure solution existence.

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.