Nonlinear Schrödinger equations with a critical, inverse-square potential

We study the existence of solutions of the following nonlinear Schrödinger equation $$-\mathrm{\Delta }u+V(x)u-\frac{(N-2{)}^2}{4\mathrm{\mid }x{\mathrm{\mid }}^2}u=f(x,u)$$ where $V:{\mathbb{R}}^N\to \mathbb{R}$ and $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ are periodic with respect to $x\in {\mathbb{R}}^N\mathrm{.}$ 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.