Quantitative profile decomposition and stability for a nonlocal Sobolev inequality

In this paper, we focus on studying the quantitative stability of the nonlocal Sobolev inequality given by$S_{HL}{\left(\underset{R^N}{\int }\right(|x{|}^{-\mu }⁎|u{|}^{2_{\mu }^{⁎}}\left)\right|u{|}^{2_{\mu }^{⁎}}dx)}^{\frac{1}{2_{\mu }^{⁎}}}\le \underset{R^N}{\int }|\nabla u{|}^{2}dx,\forall u\in D^{1,2}(R^N),$ where ⁎ denotes the convolution of functions, $2_{\mu }^{⁎}:=\frac{2N-\mu }{N-2}$ and $S_{HL}$ are positive constants that depends solely on N and μ. For $N\ge 3$ and $0<\mu <N$, it is well-known that, up to translation and scaling, the nonlocal Sobolev inequality possesses a unique extremal function $W[\xi ,\lambda ]$ that is positive and radially symmetric.

Our research consists of three main parts. Firstly, we prove a result that provides quantitative stability of the nonlocal Sobolev inequality with the level of gradients. Secondly, we establish the stability of profile decomposition in relation to the Euler-Lagrange equation of the aforementioned inequality for nonnegative functions. Lastly, we investigate the quantitative stability of the nonlocal Sobolev inequality in the following form:${\Vert \nabla u-\sum _{i=1}^{\kappa }\nabla W[{\xi }_i,{\lambda }_i]\Vert }_{L^2}\le C{\Vert \Delta u+(\frac{1}{|x{|}^{\mu }}⁎|u{|}^{2_{\mu }^{⁎}})|u{|}^{2_{\mu }^{⁎}-2}u\Vert }_{{\left(D^{1,2}\right(R^N\left)\right)}^{-1}},$ where the parameter region satisfies $\kappa \ge 2$, $3\le N<6-\mu$, $\mu \in (0,N)$ with $0<\mu \le 4$, or in the case of dimension $N\ge 3$ and $\kappa =1$, $\mu \in (0,N)$ with $0<\mu \le 4$.