Concentration phenomenon for fractional nonlinear Schr\"{o}dinger equations

TL;DR

This paper investigates the concentration behavior of solutions to fractional nonlinear Schrödinger equations, demonstrating that solutions can localize around critical points of the potential as the parameter approaches zero.

Contribution

The study applies Lyapunov-Schmidt reduction to establish concentration phenomena for fractional Schrödinger equations with nonlocal operators, extending previous results to fractional cases.

Findings

Solutions concentrate at non-degenerate critical points of V as epsilon approaches zero.

The method applies to equations with fractional Laplacians in dimensions 1 to 3.

Results cover a range of nonlinear exponents depending on s and n.

Abstract

We study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \begin{equation}\label{e:abstract} (-\varepsilon^2\Delta)^sv+Vv-|v|^{\alpha}v=0\quad\mbox{in}\quad\mathbf R^n, \end{equation} where $n =1, 2, 3$, $\max\{\frac{1}{2}, \frac{n}{4}\}< s < 1$, $1 \leq \alpha < \alpha_*(s,n)$, $V\in C^3_{b}(\mathbf{R}^n)$. Here the exponent $\alpha_*(s,n)=\frac{4s}{n-2s}$ for $0 < s < \frac{n}{2}$ and $\alpha_*(s,n)=\infty$ for $s \geq\frac{n}{2}$. Then for each non-degenerate critical point $z_0$ of $V$, there is a nontrivial solution of equation (\ref{e:abstract}) concentrating to $z_0$ as $\varepsilon\to 0$.