A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core

We consider the equation $$-\mathrm{\Delta }u+u=Q_{\epsilon }(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,u\in H^1({\mathbb{R}}^N),$$ where $Q_{\epsilon }$ takes the value $1$ on each ball $B_{\epsilon }(y)$, $y\in {\mathbb{Z}}^N$, and the value $-1$ elsewhere. We establish the existence of a least energy solution for each $\epsilon \in (0,\frac{1}{2})$ and show that their $H^1$ and $L^p$ norms concentrate locally at points of ${\mathbb{Z}}^N$ as $\epsilon \to 0$.