Concentrating standing waves for the fractional nonlinear Schr\"odinger equation

TL;DR

This paper constructs concentrating solutions for a fractional nonlinear Schrödinger equation with multiple peaks near critical points of the potential, extending classical results from the case s=1 to fractional s in (0,1).

Contribution

It generalizes known concentration results for the nonlinear Schrödinger equation to the fractional case using Lyapunov-Schmidt reduction.

Findings

Existence of multi-peak solutions near critical points of V.

Extension of classical results to fractional Laplacian case.

Solutions concentrate around nondegenerate critical points of V.

Abstract

We consider the semilinear equation $$ \epsilon^{2s} (-\Delta)^s u + V(x)u - u^p = 0, \quad u>0, \quad u\in H^{2s}(\R^N) $$ where $0<s<1,\ 1<p<\frac{N+2s}{N-2s}$, $ V(x)$ is a sufficiently smooth potential with $\inf_\R V(x)> 0$, and $\epsilon>0$ is a small number. Letting $w_\lambda$ be the radial ground state of $(-\Delta)^s w_\lambda + \lambda w_\lambda - w_\lambda^p=0$ in $H^{2s}(\R^N)$, we build solutions of the form $$ u_\epsilon(x) \sim \sum_{i=1}^k w_{\lambda_i} ((x-\xi_i^\epsilon)/\epsilon),$$ where $\lambda_i = V(\xi_i^\epsilon)$ and the $\xi_i^\epsilon $ approach suitable critical points of $V$. Via a Lyapunov Schmidt variational reduction, we recover various existence results already known for the case $s=1$. In particular such a solution exists around $k$ nondegenerate critical points of $V$. For $s=1$ this corresponds to the classical results by Floer-Weinstein and Oh.