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

TL;DR

This paper establishes the existence of solutions for a class of quasilinear Schrödinger equations with indefinite potentials using variational methods, including local linking and Morse theory, even when standard mountain pass conditions fail.

Contribution

It introduces a novel approach to handle indefinite potentials in quasilinear Schrödinger equations, extending solution existence results beyond traditional positive-definite cases.

Findings

Existence of a nontrivial solution using local linking and Morse theory.

Unbounded sequence of solutions when the nonlinearity is odd.

Handles indefinite potentials where classical mountain pass methods do not apply.

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.