Degenerate stability of critical points of the Caffarelli-Kohn-Nirenberg inequality along the Felli-Schneider curve
Yuxuan Zhou, Wenming Zou
TL;DR
The paper tackles quantitative stability for the degenerate critical Hardy-Hénon equation on the Felli-Schneider curve, where the linearized operator loses nondegeneracy. By transforming to a cylinder via the Emden-Fowler map, the authors analyze Talenti bubbles and their degeneracies using spectral theory, Lyapunov–Schmidt reduction, and higher-order expansions. They prove an optimal cubic-rate stability for the single-bubble case and extend to multi-bubble configurations under nondegeneracy-type restrictions, establishing the first rigorous degenerate stability results at critical points. The results leverage precise energy decompositions and delicate control of interaction terms, with implications for stability theory of sharp functional inequalities in degenerate regimes. The methods are expect to apply to other problems where degeneracy obstructs standard stability analyses.
Abstract
In this paper, we investigate the validity of a quantitative version of stability for the critical Hardy-Hénon equation \begin{equation*} H(u):=÷(|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\leqν}}\norm*{u-\sum_{i=1}^νU_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(ν-\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(ν+\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}. \end{equation*} Here $ν\in\N_+$ 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 $ν=1$, we obtain an optimal $F$. When $ν\geq2$ 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.