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|^{-μ} \ast |u|^{2_μ^{\ast}}\big)|u|^{2_μ^{\ast}} dx\right)^{\frac{1}{2_μ^{\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 $N\ge 3$, 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}^κ\nabla W[ξ_i,λ_i]\Big\|_{L^2}\leq C\Big\|Δu+\left(\frac{1}{|x|^μ}\ast |u|^{2_μ^{\ast}}\right)|u|^{2_μ^{\ast}-2}u\Big\|_{(\mathcal{D}^{1,2}(\mathbb{R}^N))^{-1}} \end{equation*} with the parameter region $\kappa \ge 2$, $3\le N<6-\mu$, $\mu \in (0,N)$ satisfying $0<\mu \le 4$, or dimension $N\ge 3$ and $\kappa =1$, $\mu \in (0,N)$ satisfying $0<\mu \le 4$.