Stability of Caffarelli-Kohn-Nirenberg inequality

TL;DR

This paper investigates the stability properties of the Caffarelli-Kohn-Nirenberg inequality, establishing quantitative bounds on how solutions deviate from extremal functions within certain parameter regions.

Contribution

It proves stability results for the CKN inequality in both functional and critical point frameworks, extending understanding of extremal function behavior under perturbations.

Findings

Stability in the functional inequality setting with explicit distance bounds.

Stability in the critical point setting with bounds depending on the residual term.

Results hold in specific parameter regions defined by the Felli-Schneider curve.

Abstract

In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality: \begin{eqnarray*} \bigg(\int_{{\mathbb R}^N}|x|^{-b(p+1)}|u|^{p+1}dx\bigg)^{\frac{2}{p+1}}\leq C_{a,b,N}\int_{{\mathbb R}^N}|x|^{-2a}|\nabla u|^2dx \end{eqnarray*} where $N\geq3$, $a<\frac{N-2}{2}$, $a\leq b\leq a+1$ and $p=\frac{N+2(1+a-b)}{N-2(1+a-b)}$. It is well-known that up to dilations $\tau^{\frac{N-2}{2}-a}u(\tau x)$ and scalar multiplications $Cu(x)$, the CKN inequality has a unique extremal function $W(x)$ which is positive and radially symmetric in the parameter region $b_{FS}(a)\leq b<a+1$ with $a<0$ and $a\leq b<a+1$ with $a\geq0$ and $a+b>0$, where $b_{FS}(a)$ is the Felli-Schneider curve. We prove that in the above parameter region the following stabilities hold: \begin{enumerate} \item[$(1)$] \quad stability of CKN inequality in the functional inequality setting $$dist_{D^{1,2}_{a}}^2(u, \mathcal{Z})\lesssim\|u\|^2_{D^{1,2}_a({\mathbb R}^N)}-C_{a,b,N}^{-1}\|u\|^2_{L^{p+1}(|x|^{-b(p+1)},{\mathbb R}^N)}$$ where $\mathcal{Z}= \{ c W_\tau\mid c\in\bbr\backslash\{0\}, \tau>0\}$; \item[$(2)$]\quad stability of CKN inequality in the critical point setting (in the class of nonnegative functions) \begin{eqnarray*} dist_{D_a^{1,2}}(u, \mathcal{Z}_0^\nu)\lesssim\left\{\aligned &\Gamma(u),\quad p>2\text{ or }\nu=1,\\ &\Gamma(u)|\log\Gamma(u)|^{\frac12},\quad p=2\text{ and }\nu\geq2,\\ &\Gamma(u)^{\frac{p}{2}},\quad 1<p<2\text{ and }\nu\geq2, \endaligned\right. \end{eqnarray*} where $\Gamma (u)=\|div(|x|^{-a}\nabla u)+|x|^{-b(p+1)}|u|^{p-1}u\|_{(D^{1,2}_a)^{'}}$ and $$\mathcal{Z}_0^\nu=\{(W_{\tau_1},W_{\tau_2},\cdots,W_{\tau_\nu})\mid \tau_i>0\}.$$