Stability estimates for critical points of a nonlocal Sobolev-type inequality
Minbo Yang, Shunneng Zhao
TL;DR
This work proves a sharp quantitative stability result for the nonlocal Sobolev-type inequality $C_{HLS}(\int(|x|^{-\mu}*u^{p})u^{p})^{1/p} \le \int|\nabla u|^{2}$ with $p=(2n-\mu)/(n-2)$ in dimensions $n\ge3$ and $0<\mu<n$. Building on nonlocal profile decompositions, the authors develop a weighted Lyapunov-Schmidt framework around sums of Aubin-Talenti-type bubbles $W[\xi_i,\lambda_i]$, deriving detailed convolution estimates for Hartree-type interactions and establishing robust a priori bounds in a carefully designed weighted space. The main result shows a quantitative bound on the distance of a near-extremal function to the manifold of multi-bubble configurations, specifically dist$_{D^{1,2}}(u,\mathcal{M}_0)\le C\,\tau_{n,\mu}(\|\hat f\|_{(D^{1,2})^{-1}})$, with explicit control of bubble interactions through integrals of convolution terms. This extends local stability results to the nonlocal Hartree setting in the regime $n\ge6-\mu$ and $\mu\in(0,4]$, and leverages a nondegeneracy framework to achieve the three-to-five sentence high-level impact: it advances understanding of stability for nonlocal critical problems, provides a rigorous pathway for profile decompositions in nonlocal inequalities, and supplies tools potentially applicable to Hartree-type PDEs and related nonlocal variational problems.
Abstract
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|^{-μ} \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 $p=\frac{2n-μ}{n-2}$, $n\geq3$ and $μ\in(0,n)$, is energy-critical exponent and $C_{HLS}$ is the best constant depending on $n$ and $μ$. 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*} Δu+(|x|^{-μ}\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 $κ\geq2$ if dimension $3\leq n<6-μ$ and $μ\in(0,n)$ with $μ\in(0,4]$ 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 $n\geq6-μ$ and $μ\in(0,4)$, which is an extension of the recent works by Deng-Sun-Wei in \cite{DSW21} for the classical Sobolev inequality.