Quantitative stability of a nonlocal Sobolev inequality

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 $0<\mu\leq4$.