Stability of the Caffarelli-Kohn-Nirenberg inequality along Felli-Schneider curve: critical points at infinity

Abstract

In this paper, we consider the following Caffarelli-Kohn-Nirenberg (CKN for short) inequality \begin{eqnarray*} \bigg(\int_{{\mathbb R}^d}|x|^{-b(p+1)}|u|^{p+1}dx\bigg)^{\frac{2}{p+1}}\leq S_{a,b}\int_{{\mathbb R}^d}|x|^{-2a}|\nabla u|^2dx, \end{eqnarray*} where $u\in D^{1,2}_{a}({\mathbb R}^d)$, $d\geq2$, $p=\frac{d+2(1+a-b)}{d-2(1+a-b)}$ and \begin{eqnarray}\label{eq0003} \left\{\aligned &a<b<a+1,\quad d=2,\\ &a\leq b<a+1,\quad d\geq3. \endaligned \right. \end{eqnarray} Based on the ideas of \cite{DSW2024,FP2024}, we develop a suitable strategy to derive the following sharp stability of the critical points at infinity of the above CKN inequality in the degenerate case $d\geq2$, $b=b_{FS}(a)$ (Felli-Schneider curve) and $a<0$: let $\nu \in {\mathbb N}$ and $u\in D^{1,2}_{a}({\mathbb R}^d)$ be an nonnegative function such that \begin{eqnarray}\label{eqqqnew0001} \left(\nu-\frac12\right)\left(S_{a,b}^{-1}\right)^{\frac{p+1}{p-1}}<\|u\|^2_{D^{1,2}_a({\mathbb R}^d)}<\left(\nu+\frac12\right)\left(S_{a,b}^{-1}\right)^{\frac{p+1}{p-1}} \end{eqnarray} Then we have the following sharp inequality \begin{eqnarray*} \inf_{\overrightarrow{\alpha}_{\nu}\in\left({\mathbb R}_+\right)^{\nu}, \overrightarrow{\lambda}_{\nu}\in {\mathbb R}^\nu}\left\|u-\sum_{j=1}^{\nu}\alpha_j W_{\lambda_j}\right\|\lesssim\left\|-div(|x|^{-a}\nabla u)-|x|^{-b(p+1)}|u|^{p-1}u\right\|_{W^{-1,2}_a({\mathbb R}^d)}^{\frac{1}{3}} \end{eqnarray*} as $\left\|-div(|x|^{-a}\nabla u)-|x|^{-b(p+1)}|u|^{p-1}u\right\|_{W^{-1,2}_a({\mathbb R}^d)}\to0$. The significant finding in our result is that in the degenerate case, {\it the power of the optimal stability is an absolute constant $1/3$} (independent of $p$ and $\nu$) which is quite different from the non-degenerate case \cite{DSW2024,WW2022}.