Ground states of a system of nonlinear Schr\"odinger equations with periodic potentials

TL;DR

This paper establishes the existence of ground state solutions for a system of coupled nonlinear Schrödinger equations with periodic potentials, using a novel linking approach on the Nehari-Pankov manifold.

Contribution

It introduces a new linking-type variational method on the Nehari-Pankov manifold to find ground states for coupled Schrödinger systems with periodic potentials.

Findings

Existence of ground state solutions under general conditions.

Application of a new linking technique involving the Nehari-Pankov manifold.

Solutions minimize the associated energy functional.

Abstract

We are concerned with a system of coupled Schr\"odinger equations $$-\Delta u_i + V_i(x)u_i = \partial_{u_i}F(x,u)\hbox{ on }\mathbb{R}^N,\,i=1,2,...,K,$$ where $F$ and $V_i$ are periodic in $x$ and $0\notin \sigma(-\Delta+V_i)$ for $i=1,2,...,K$, where $\sigma(-\Delta+V_i)$ stands for the spectrum of the Schr\"odinger operator $-\Delta+V_i$. We impose general assumptions on the nonlinearity $F$ with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with the system on a Nehari-Pankov manifold. Our approach is based on a new linking-type result involving the Nehari-Pankov manifold.