Ground States of K-component Coupled Nonlinear Schrödinger Equations with Inverse-square Potential

Abstract

In this paper, the authors study ground states for a class of K-component coupled nonlinear Schrödinger equations with a sign-changing potential which is periodic or asymptotically periodic. The resulting problem engages three major difficulties: One is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the ℤ ^N-translation invariance, many effective methods for periodic problems cannot be applied to asymptotically periodic ones. The third difficulty is singular potential ${{{\mu _i}} \over {{{\left| x \right|}^2}}}$, which does not belong to the Kato’s class. These enable them to develop a direct approach and new tricks to overcome the difficulties caused by singularity and the dropping of periodicity of potential.

Keywords

Schrödinger equations / Ground states / Strongly indefinite functionals / Non-Nehari manifold method

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