Systems of coupled Schr\"odinger equations with sign-changing nonlinearities via classical Nehari manifold approach

TL;DR

This paper establishes existence and multiplicity of solutions for coupled Schr"odinger systems with sign-changing nonlinearities using variational methods, including the Nehari manifold and Fountain theorem, in bounded and unbounded domains.

Contribution

It introduces a classical Nehari manifold approach combined with variational techniques to handle sign-changing nonlinearities in Schr"odinger systems, extending results to unbounded domains with periodic potentials.

Findings

Existence of solutions in bounded domains via Nehari manifold and Fountain theorem.

Multiplicity results for solutions in unbounded periodic domains.

Application of concentration-compactness and Lusternik-Schnirelmann methods.

Abstract

We propose existence and multiplicity results for the system of Schr\"odinger equations with sign-changing nonlinearities in bounded domains or in the whole space $\mathbb{R}^N$. In the bounded domain we utilize the classical approach via the Nehari manifold, which is (under our assumptions) a differentiable manifold of class $\mathcal{C}^1$ and the Fountain theorem by Bartsch. In the space $\mathbb{R}^N$ we additionally need to assume the $\mathbb{Z}^N$-periodicity of potentials and our proofs are based on the concentration-compactness lemma by Lions and the Lusternik-Schnirelmann values.