Stability of Hardy-Sobolev Inequality

Given N≥ 3, we consider the critical Hardy-Sobolev equation -Δ u-γ/|x|²u=|u|2^*(s)-2u|x|ˢ in RN \0\, where 0<γ<γH:=(N-2/2)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) Γ(u) in the spirit of Ciraolo-Figalli-Maggi (2017).