Standing waves for quasilinear Schr\"{o}inger equations with indefinite potentials

Abstract

We consider quasilinear Schr\"{o}dinger equations in $\mathbb{R}^{N}$ of the form% \[ -\Delta u+V(x)u-u\Delta(u^{2})=g(u)\text{,}% \] where $g(u)$ is $4$-superlinear. Unlike all known results in the literature, the Schr\"{o}dinger operator $-\Delta+V$ is allowed to be indefinite, hence the variational functional does not satisfy the mountain pass geometry. By a local linking argument and Morse theory, we obtain a nontrivial solution for the problem. In case that $g$ is odd, we get an unbounded sequence of solutions.