Stability estimates for critical points of a nonlocal Sobolev-type inequality

In this paper, we study the stability of the following nonlocal Soblev-type inequality \begin{equation*} C_{HLS}\big(\int_{\mathbb{R}^n}\big(|x|^{-\mu} \ast u^{p}\big)u^{p} dx\big)^{\frac{1}{p}}\leq\int_{\mathbb{R}^n}|\nabla u|^2 dx , \quad \forall~u\in D^{1,2}(\mathbb{R}^n), \end{equation*} which is induced by the classical Sobolev inequality and the Hardy-Littlewood-Sobolev inequality, where $p=\frac{2n-\mu}{n-2}$, $n\geq3$ and $\mu\in(0,n)$, is energy-critical exponent and $C_{HLS}$ is the best constant depending on $n$ and $\mu$. Up to translation and scaling, the best constant of the nonlocal Soblev inequality can be achieved by a unique family of positive and radially symmetric extremal function $W(x)$ that satisfies, up to a suitable scaling, the classical critical Hartree equation \begin{equation*} \Delta u+(|x|^{-\mu}\ast u^{p})u^{p-1}=0 \quad \mbox{in}\quad \mathbb{R}^n. \end{equation*} Recently, Piccione, Yang and Zhao in \cite{p-y-z24} established a nonlocal version of Struwe's profile decomposition and they only proved the nonlocal version of the quantitative stability for the one bubble case without dimension restriction and the multiple bubbles case $\kappa\geq2$ if dimension $3\leq n<6-\mu$ and $\mu\in(0,n)$ with $\mu\in(0,4]$ in Ciraolo-Figalli-Maggi \cite{CFM18} and Figalli-Glaudo \cite{FG20}. We establish the quantitative stability estimates for critical point of the nonlocal Soblev inequality for $n\geq6-\mu$ and $\mu\in(0,4)$, which is an extension of the recent works by Deng-Sun-Wei in \cite{DSW21} for the classical Sobolev inequality.