On the existence of positive solutions for a quasilinear Schr\"{o}dinger equation

TL;DR

This paper proves the existence of positive solutions for a quasilinear Schrödinger equation under new conditions that do not require the nonlinearity to be 4-superlinear at infinity, expanding the understanding of such equations.

Contribution

It introduces a novel approach to establish positive solutions without assuming 4-superlinearity of the nonlinearity at infinity.

Findings

Existence of positive solutions under new assumptions on V and h.

No need for h to be 4-superlinear at infinity.

Method applicable to a broader class of quasilinear Schrödinger equations.

Abstract

This paper is concerned with the quasilinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u- \Delta(u^2)u =h(u), \ \ \mbox{in} \ \mathbb{R}^N, \end{equation*} where $N\geq 3$. Under appropriate assumptions on $V$ and $h$, we establish the existence of positive solutions. The main novelty is that, unlike most other papers on this problem, we do not assume $h$ is 4-superlinear at infinity.