Multi-peak solutions for the fractional Schr\"odinger equation with Dirichlet datum

TL;DR

This paper studies the existence of multi-peak solutions for a fractional Schrödinger equation with Dirichlet boundary conditions, showing how solutions concentrate at critical points of the potential as a parameter tends to zero.

Contribution

It constructs positive multi-peak solutions concentrating at prescribed non-degenerate critical points of the potential for the fractional Schrödinger equation.

Findings

Solutions concentrate at prescribed critical points of V.

Existence of multi-peak solutions depends on the nature of critical points.

Clustering phenomena occur around local maxima but not minima.

Abstract

Let $s\in (0,1)$, $\varepsilon>0$ and let $\Omega$ be a bounded smooth domain. Given the problem $$\varepsilon^{2s}(-\Delta)^{s} u + V(x)u = |u|^{p-1}u \quad \mbox{in }\; \Omega,$$ with Dirichlet boundary conditions and $1<p<(n+2s)/(n-2s)$, we analyze the existence of positive multi-peak solutions concentrating, as $\varepsilon\to 0$, to one or several points of $\Omega$. Under suitable conditions on $V$, we construct positive solutions concentrating at any prescribed set of its non degenerate critical points. Furthermore, we prove existence and non existence of clustering phenomena around local maxima and minima of $V$, respectively. The proofs rely on a Lyapunov-Schmidt reduction where three effects need to be controlled: the potential, the boundary and the interaction among peaks. The slow decay of the associated {\it ground-state} demands very precise asymptotic expansions.