Systems of coupled Schrödinger equations with sign-changing nonlinearities via classical Nehari manifold approach

Abstract

We propose existence and multiplicity results for the system of Schrödinger 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.