Degenerate stability of critical points of the Caffarelli-Kohn-Nirenberg inequality along the Felli-Schneider curve

Abstract

In this paper, we investigate the validity of a quantitative version of stability for the critical Hardy-H\'enon equation \begin{equation*} H(u):=\div(|x|^{-2a}\nabla u)+|x|^{-pb}|u|^{p-2}u=0,\quad u\in D_a^{1,2}(\R^n), \end{equation*} \begin{equation*} n\geq 2,\quad a<b<a+1,\quad a<\frac{n-2}{2},\quad p=\frac{2n}{n-2+2(b-a)}, \end{equation*} which is well known as the Euler-Lagrange equation of the classical Caffarelli-Kohn-Nirenberg inequality. Establishing quantitative stability for this equation amounts to finding a nonnegative function $F$ such that the estimate \begin{equation*} \inf_{\substack{U_i\in\mathcal{M} 1\leq i\leq\nu}}\norm*{u-\sum_{i=1}^\nu U_i}_{D_a^{1,2}(\R^n)}\leq C(a,b,n)F(\norm*{H(u)}_{D_a^{-1,2}(\R^n)}) \end{equation*} holds for any nonnegative function $u$ satisfying \begin{equation*} \left(\nu-\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}\leq\int_{\R^n}|x|^{-2a}|\nabla u|^2\mathrm{d}x\leq \left(\nu+\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}. \end{equation*} Here $\nu\in\N_+$ and $\mathcal{M}$ denotes the set of positive solutions to this equation. When $(a,b)$ falls above the Felli-Schneider curve, Wei and Wu \cite{Wei} found an optimal $F$. Their proof relies heavily on the fact that $\mathcal{M}$ is non-degenerate. When $(a,b)$ falls on the Felli-Schneider curve, due to the absence of the non-degeneracy condition, it becomes complicated and technical to find a suitable $F$. In this paper, we focus on this case. When $\nu=1$, we obtain an optimal $F$. When $\nu\geq2$ and $u$ is not too degenerate, we also derive an optimal $F$. To our knowledge, the results in this paper provide the first instance of degenerate stability in the critical point setting. We believe that our methods will be useful in other works on degenerate stability.