Stability of Hardy-Sobolev Inequality

Given $N\ge 3,$ we consider the critical Hardy-Sobolev equation $-\mathrm{\Delta }u-\frac{\gamma }{\mathrm{\mid }x{\mathrm{\mid }}^2}u=\frac{\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }(s)-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^s}$ in ${\mathbb{R}}^N\setminus \{0\},$ where $0<\gamma <{\gamma }_H:={\left(\frac{N-2}{2}\right)}^2,s\in (0,2)$ and $2^{\ast }(s)=\frac{2(N-s)}{(N-2)}\mathrm{.}$ 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 $\mathrm{dist}(u,\mathcal{M})\lesssim \mathrm{\Gamma }(u)$ in the spirit of Ciraolo-Figalli-Maggi (2017).