Uniqueness and nondegeneracy of solutions for a critical nonlocal equation

TL;DR

This paper classifies positive solutions of a critical nonlocal equation involving the Riesz potential, proving their symmetry, uniqueness, and nondegeneracy for certain parameter ranges using the moving plane method.

Contribution

It establishes the symmetry, uniqueness, and nondegeneracy of positive solutions for a class of critical nonlocal equations, extending understanding of their structure.

Findings

Positive solutions are symmetric and unique.

Solutions are nondegenerate when lose to N.

The moving plane method in integral form is effective for analysis.

Abstract

The aim of this paper is to classify the positive solutions of the nonlocal critical equation: $$ -\Delta u=\left(I_{\mu}\ast u^{2^{\ast}_{\mu}}\right)u^{{2}^{\ast}_{\mu}-1},~~x\in\mathbb{R}^{N}, $$ where $0<\mu<N$, if $N=3\ \hbox{or} \ 4$ and $0<\mu\leq4$ if $N\geq5$, $I_{\mu}$ is the Riesz potential defined by $$ I_{\mu}(x)=\frac{\Gamma(\frac{\mu}{2})}{\Gamma(\frac{N-\mu}{2})\pi^{\frac{N}{2}}2^{N-\mu}|x|^{\mu}} $$ with $\Gamma(s)=\int^{+\infty}_{0}x^{s-1}e^{-x}dx$, $s>0$ and $2^{\ast}_{\mu}=\frac{2N-\mu}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We apply the moving plane method in integral forms to prove the symmetry and uniqueness of the positive solutions and prove the nondegeneracy of the unique solutions for the equation when $\mu$ close to $N$.