Quantitative stability of a nonlocal Sobolev inequality

TL;DR

This paper investigates the stability of the nonlocal Sobolev inequality, establishing quantitative stability results, profile decomposition stability, and analyzing the inequality's stability in various parameter regimes.

Contribution

It provides the first quantitative stability results for the nonlocal Sobolev inequality, including gradient level stability and profile decomposition stability for the associated Euler-Lagrange equation.

Findings

Proved quantitative stability of the nonlocal Sobolev inequality.

Established stability of profile decomposition for the Euler-Lagrange equation.

Analyzed stability of the inequality under various parameter conditions.

Abstract

In this paper, we study the quantitative stability of the nonlocal Soblev inequality \begin{equation*} S_{HL}\left(\int_{\mathbb{R}^N}\big(|x|^{-\mu} \ast |u|^{2_{\mu}^{\ast}}\big)|u|^{2_{\mu}^{\ast}} dx\right)^{\frac{1}{2_{\mu}^{\ast}}}\leq\int_{\mathbb{R}^N}|\nabla u|^2 dx , \quad \forall~u\in \mathcal{D}^{1,2}(\mathbb{R}^N), \end{equation*} where $2_{\mu}^{\ast}=\frac{2N-\mu}{N-2}$ and $S_{HL}$ is a positive constant depending only on $N$ and $\mu$. For $N\geq3$, and $0<\mu<N$, it is well-known that, up to translation and scaling, the nonlocal Soblev inequality has a unique extremal function $W[\xi,\lambda]$ which is positive and radially symmetric. We first prove a result of quantitative stability of the nonlocal Soblev inequality with the level of gradients. Secondly, we also establish the stability of profile decomposition to the Euler-Lagrange equation of the above inequality for nonnegative functions. Finally we study the stability of the nonlocal Soblev inequality \begin{equation*} \Big\|\nabla u-\sum_{i=1}^{\kappa}\nabla W[\xi_i,\lambda_i]\Big\|_{L^2}\leq C\Big\|\Delta u+\left(\frac{1}{|x|^{\mu}}\ast |u|^{2_{\mu}^{\ast}}\right)|u|^{2_{\mu}^{\ast}-2}u\Big\|_{(\mathcal{D}^{1,2}(\mathbb{R}^N))^{-1}} \end{equation*} with the parameter region $\kappa\geq2$, $3\leq N<6-\mu$, $\mu\in(0,N)$ satisfying $0<\mu\leq4$, or dimension $N\geq3$ and $\kappa=1$, $\mu\in(0,N)$ satisfying $0<\mu\leq4$.