Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality

In this article, we prove the best Bianchi-Egnell constant for the Hardy-Sobolev (HS) inequality {align*} C_{}(γ) := infu \ not an optimizer {∫R^n (|∇ u|^2 - γ{|x|^2}u^2) \ { d}x - S_γ\|u\|_{L^{2^{}}}^2}{dist (u, \ set of optimizers)^2}, {align*} is attained, extending the result of König [arXiv:2211.14185] for the classical Sobolev inequality (that corresponds to $γ= 0$). One of the main difficulties is that the third eigenspace of the linearized operator may contain only spherical harmonics of degree $1$, and hence, an essential non-vanishing criterion fails [arXiv:2210.08482]. This non-vanishing criterion is indispensable for proving the best Bianchi-Egnell constant C_(γ) < C_^(γ) that prevents a minimizing sequence converging to one of the optimizers. In addition, not being translation invariant, extracting a non-zero weak limit from a minimizing sequence presents difficulties. We found another hidden critical level C_(γ) <1 - S_γ/S, where S is the best Sobolev constant that plays a significant role in proving the existence of an extremizer. In particular, we show that there exists a γ₀>0 such that for γ≥ γ₀,\ C_(γ) is attained. Moreover, we remark that there is a region γ₀ ≤ γ< γc^, where the third eigenspace of the linearized operator contains only spherical harmonics of degree $1.$ Our result improves some of the results in Wei-Wu [arXiv:2308.04667] corresponding to the HS inequality.