Solutions to a nonlinear Schr\"odinger equation with periodic potential and zero on the boundary of the spectrum

TL;DR

This paper investigates solutions to a nonlinear Schrödinger equation with periodic potential, establishing the existence of a ground state and infinitely many solutions under certain symmetry conditions using Nehari manifold techniques.

Contribution

It introduces a novel application of Nehari manifold methods to a strongly indefinite nonlinear Schrödinger equation with periodic potential, proving existence and multiplicity of solutions.

Findings

Existence of a ground state solution was proven.

Infinitely many solutions exist if g is odd.

Solutions are geometrically distinct under symmetry conditions.

Abstract

We study the following nonlinear Schr\"odinger equation $$-\Delta u + V(x) u = g(x,u),$$ where V and g are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $-\Delta+V$. The superlinear and subcritical term g satisfies a Nehari type monotonicity condition. We employ a Nehari manifold type technique in a strongly indefitnite setting and obtain the existence of a ground state solution. Moreover we get infinitely many geometrically distinct solutions provided that g is odd.