Schr\"{o}dinger-type equations with sign-changing nonlinearities: a survey

TL;DR

This survey reviews recent variational methods, especially the Nehari manifold technique, for finding solutions to nonlinear Schr"odinger-type equations with sign-changing nonlinearities and various potentials.

Contribution

It provides an introduction to the problem and summarizes recent advances in establishing the existence of ground and bound states using variational approaches.

Findings

Survey of variational methods for sign-changing nonlinearities

Summary of existence results for ground states

Overview of techniques for unbounded or singular potentials

Abstract

We are looking for solutions to nonlinear Schr\"odinger-type equations of the form $$ (-\Delta)^{\alpha / 2} u (x) + V(x) u(x) = h (x,u(x)), \quad x \in \mathbb{R}^N, $$ where $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is an external potential (bounded, unbounded or singular) and the nonlinearity $h : \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}$ may be sign-changing. The aim of this paper is to give an introduction to the problem and the variational approach based on the Nehari manifold technique, and to survey recent results concerning the existence of ground and bound states.