On the sharp quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality in ${\mathbb{R}}^n$ with $n\ge 3$

Abstract

Assume $n\ge 3$ and $u\in {\dot{H}}^1({\mathbb{R}}^n)$. Recently, Piccione, Yang and Zhao \cite{Piccione-Yang-Zhao} established a nonlocal version of Struwe's decomposition in \cite{Struwe-1984}, i.e., if $\Gamma (u):={\Vert \Delta u+D_{n,\alpha }{\int }_{{\mathbb{R}}^n}\frac{|u{|}^{p_{\alpha }}(y)}{|x-y{|}^{\alpha }}\mathrm{d}y|u{|}^{p_{\alpha }-2}u\Vert }_{{\dot{H}}^{-1}}\to 0$ and $u\ge 0$, then $dist(u,\mathcal{T})\to 0$, where $dist(u,\mathcal{T})$ denotes the ${\dot{H}}^1({\mathbb{R}}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. In this paper, we establish the nonlocal version of the quantitative estimates of Struwe's decomposition in Ciraolo, Figalli and Maggi \cite{CFM} for one bubble and $n\ge 3$, Figalli and Glaudo \cite{Figalli-Glaudo2020} for $3\le n\le 5$ and Deng, Sun and Wei \cite{DSW} for $n\ge 6$ and two or more bubbles. We prove that for $n\ge 3$ and $0<\alpha <n$, \[dist (u,\mathcal{T})\leq C\begin{cases} Γ(u)\left|\log Γ(u)\right|^{\frac{1}{2}}\quad&\text{if } \,\, n\geq 6, \,\, ν\geq2 \,\, \text{and} \,\, α=\frac{n+2}{2}, \\ Γ(u) \quad&\text{for any other cases,}\end{cases}\] where $\nu$ denotes the number of bubbles. Furthermore, we show that this inequality is sharp for $n\ge 6$ and $\alpha =\frac{n+2}{2}$. It should be emphasized that, in our paper, we have developed new techniques to deal with the strong singular case $4<\alpha <n$, which can not be handled by reduction methods in previous works. We believe that our method can also be applied to other problems related to the physically interesting Hartree equation.

We corrected a few typos in the manuscript. This paper has been accepted for publication in Mathematische Annalen