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}}}(\gamma) := \inf_{{u \ \small \mbox{not an optimizer}}} \frac{\int_{\mathbb{R}^n} \left(|\nabla u|^2 - \frac{\gamma}{|x|^2}u^2\right) \ {\rm d}x - S_{\gamma}\|u\|_{L^{2^{\star}}}^2}{\mbox{dist} (u, \ \mbox{set of optimizers})^2}, \end{align*} is attained, extending the result of K\"onig [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 $C_{\tiny\mbox{{BE}}}(\gamma) < C_{\tiny\mbox{{BE}}}^{\tiny\mbox{{loc}}}(\gamma)$ 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 $C_{\tiny\mbox{{BE}}}(\gamma) <1 - \frac{S_{\gamma}}{S},$ 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 $\gamma \geq \gamma_0,\ C_{\tiny\mbox{{BE}}}(\gamma)$ is attained. Moreover, we remark that there is a region $\gamma_0 \leq \gamma < \gamma_c^{\star},$ 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.