Ground states of nonlinear Schr\"odinger equations with sum of periodic and inverse-square potentials

We study the existence of solutions of the following nonlinear Schr\"odinger equation \begin{equation*} -\Delta u + \Big(V(x)-\frac{\mu}{|x|^2}\Big) u = f(x,u) \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathrm{R}^N\times\mathbb{R}\to\mathbb{R}$ are periodic in $x\in\mathbb{R}$. We assume that $0$ does not lie in the spectrum of $-\Delta+V$ and $\mu<\frac{(N-2)^2}{4}$, $N\geq 3$. The superlinear and subcritical term $f$ satisfies a weak monotonicity condition. For sufficiently small $\mu\geq 0$ we find a ground state solution as a minimizer of the energy functional on a natural constraint. If $\mu<0$ and $0$ lies below the spectrum of $-\Delta+V$, then ground state solutions do not exist.