Stability of Hardy-Sobolev Inequality

Abstract

Given $N\ge 3,$ we consider the critical Hardy-Sobolev equation $-\Delta u-\frac{\gamma }{|x{|}^2}u=\frac{|u{|}^{2^{\ast }(s)-2}u}{|x{|}^s}$ in ${\mathbb{R}}^N\setminus \{0\},$ where $0<\gamma <{\gamma }_H:={\left(\frac{N-2}{2}\right)}^2,s\in (0,2)$ and $2^{\ast }(s)=\frac{2(N-s)}{(N-2)}.$ We prove a stability estimate for the corresponding Hardy-Sobolev inequality in the spirit of Bianchi-Egnell (1991). Also, we obtain a Struwe-type decomposition (1984) for the corresponding Euler-Lagrange equation. Finally, we prove a quantitative bound for one bubble, namely $\mathrm{dist}(u,\mathcal{M})\lesssim \Gamma (u)$ in the spirit of Ciraolo-Figalli-Maggi (2017).

arXiv comment24 pages