Nonlinear Schrödinger equations with a critical, inverse-square potential

Abstract

We study the existence of solutions of the following nonlinear Schrödinger equation $$-\hat{I}”u+V(x)u-\frac{(N-2{)}^2}{4|x{|}^2}u=f(x,u)$$ where $V:{\mathbb{R}}^N\to \mathbb{R}$ and $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ are periodic with respect to $x\in {\mathbb{R}}^N.$ We assume that $V$ has positive essential infimum, $f$ satisfies weak growth conditions and $N\ge 3$. The approach to the problem uses variational methods with nonstandard functional setting. We obtain the existence of the ground state solution using the new profile decomposition.