A concentration phenomenon for a semilinear Schr\"odinger equation with periodic self-focusing core

TL;DR

This paper studies a semilinear Schrödinger equation with a periodic pattern of focusing and defocusing regions, proving the existence of least energy solutions that concentrate at lattice points as the pattern scale shrinks.

Contribution

It establishes the existence of least energy solutions for the equation with a periodic coefficient and describes their concentration behavior as the period tends to zero.

Findings

Existence of least energy solutions for each small psilon

Solutions concentrate at lattice points psilon as psilon

Solutions' norms localize near psilon points

Abstract

We consider the equation $$-\Delta u+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ 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 $\varepsilon\to 0$.