Soliton solutions for a class of quasilinear Schr\"{o}dinger equations with a parameter

Abstract

Using variational methods combined with perturbation arguments, we study the existence of nontrivial classical solution for the quasilinear Schr\"{o}dinger equation \begin{equation*}\label{1.1} -\Delta u+ V(x)u+ \frac{\kappa}{2}[\Delta |u|^2]u=l(u),\ x\in\mathbb{R}^N, \end{equation*} where $V:\mathbb{R}^N\rightarrow \mathbb{R}$ and $l:\mathbb{R} \to \mathbb{R}$ are continuous function, $\kappa$ is a parameter and $N\geq 3$. This model has been proposed in plasma physics and nonlinear optics. As a main novelty with respect to some previous results, we are able to deal with the case $\kappa>0$.