Standing waves for quasilinear Schröinger equations with indefinite potentials

Abstract

We consider quasilinear Schrödinger equations in ${\mathbb{R}}^N$ of the form% \[ -Δu+V(x)u-uΔ(u^{2})=g(u)\text{,}% \] where $g(u)$ is $4$-superlinear. Unlike all known results in the literature, the Schrödinger 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.

References in corpus (1)

• Electrons on Hexagonal lattices and applications to nanotubes