Equation with positive coefficient in the quasilinear term and vanishing potential

Abstract

In this paper we study the existence of nontrivial classical solution for the quasilinear Schrödinger equation: $$ - \Delta u +V(x)u+\frac{\kappa}{2}\Delta (u^{2})u= f(u), $$ in $\mathbb{R}^N$, where $N\geq 3$, $f$ has subcritical growth and $V$ is a nonnegative potential. For this purpose, we use variational methods combined with perturbation arguments, penalization technics of Del Pino and Felmer and Moser iteration. As a main novelty with respect to some previous results, in our work we are able to deal with the case $\kappa \ge 0$ and the potential can vanish at infinity.