Stability of Hardy-Sobolev Inequality

Given N 3, we consider the critical Hardy-Sobolev equation - u-|x|²u=|u|^2^* (s) -2u|x|ˢ in RN \0\, where 0<<₇: = (N-22) ², \, s (0, 2) and 2^* (s) =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 dist (u, M) (u) in the spirit of Ciraolo-Figalli-Maggi (2017).