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 \begin{align*} C_{\tiny\mbox{BE}}(γ) := \inf_{u \ \small \mbox{not an optimizer}} \frac{\int_{$\mathbb{R}$^n} \left(|\nabla u|^2 - \fracγ{|x|^2}u^2\right) \ {\rm d}x - S_γ|u|_{L^{2^{\star}}}^2}{\mbox{dist} (u, \ \mbox{set of optimizers})^2}, \end{align*} is attained, extending the result of König [arXiv:2211.14185] for the classical Sobolev inequality (that corresponds to $\gamma =0$). 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 $C_{\text{tinyBE}}(\gamma )<C_{\text{tinyBE}}^{\text{tinyloc}}(\gamma )$ 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 $C_{\text{tinyBE}}(\gamma )<1-\frac{S_{\gamma }}{S},$ 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 ${\gamma }_0>0$ such that for $\gamma \ge {\gamma }_0,C_{\text{tinyBE}}(\gamma )$ is attained. Moreover, we remark that there is a region ${\gamma }_0\le \gamma <{\gamma }_c^{\star },$ 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.