Existence, non-degeneracy and local uniqueness of multi-peak solutions to the fractional Schr\"odinger equation with nearly critical exponent in $\mathbb{R}^N$

TL;DR

This paper constructs multi-peak solutions for a fractional Schrödinger equation with nearly critical exponent, proving their non-degeneracy and local uniqueness using Lyapunov-Schmidt reduction and Pohozaev identities.

Contribution

It introduces a novel analysis of multi-peak solutions for fractional Schrödinger equations with critical exponents, including non-degeneracy and uniqueness results.

Findings

Constructed multi-peak solutions using Lyapunov-Schmidt reduction.

Proved non-degeneracy and local uniqueness for solutions with 1/2<s<1.

Developed new techniques for local Pohozaev identities for fractional Laplacian.

Abstract

In this paper, we consider the following fractional Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{lcl} (-\Delta)^{s}u+V(x)u=u^{{p_s}-\epsilon}\ \ \ &\hbox{in}\ \mathbb{R}^N,\\ u>0\ \ \ &\hbox{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $0<s<1$, $\epsilon>0$, $p_s=(N+2s)/(N-2s)$, $N>4s$ and $V(x)\in C^1(\mathbb{R}^N)\cap L^\infty (\mathbb{R}^N)$ is non-negative. We first use the Lyapunov-Schmidt reduction method to construct multi-peak solutions to the above equation provided that $V(x)$ possesses $k$ stable critical points. Then we prove the non-degeneracy and local uniqueness of the multi-peak solutions, for $\frac{1}{2}<s<1$, $N\geq 6s$, via the blow-up argument based on various local Pohozaev identities. Due to the nonlocal property of the fractional Laplacian, we need to make delicate analysis of the approximate solutions and establish the local Pohozaev identities for the corresponding harmonic extension instead of $u$. This approach not only requires to develop refined estimates for several integrals in the local Pohozaev identities, but also to apply Pohozaev identities through a markedly different way.