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

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