Nonlinear Schr\"odinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities

TL;DR

This paper investigates the existence of ground state solutions for a nonlinear Schrödinger equation with a combined periodic and localized potential, allowing sign-changing nonlinearities and employing a Nehari manifold approach without monotonicity conditions.

Contribution

It introduces a novel analysis for ground states in Schrödinger equations with mixed potentials and sign-changing nonlinearities, extending variational methods beyond traditional monotonicity assumptions.

Findings

Established existence of ground state solutions under new conditions.

Extended variational methods to handle sign-changing nonlinearities.

Demonstrated minimization on the Nehari manifold without monotonicity.

Abstract

We look for ground state solutions to the following nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic potential $V_{per}$ and a localized potential $V_{loc}$, $\Gamma\in L^{\infty}(\mathbb{R}^N)$ is periodic and $\Gamma(x)\geq 0$ for a.e. $x\in\mathbb{R}^N$ and $2\leq q<2^*$. We assume that $\inf\sigma(-\Delta+V)>0$, where $\sigma(-\Delta+V)$ stands for the spectrum of $-\Delta +V$ and $f$ has the subcritical growth but higher than $\Gamma(x)|u|^{q-2}u$, however the nonlinearity $f(x,u)-\Gamma(x)|u|^{q-2}u$ may change sign. Although a Nehari-type monotonicity condition for the nonlinearity is not satisfied we investigate the existence of ground state solutions being minimizers on the Nehari manifold.