On the stability of the Caffarelli–Kohn–Nirenberg inequality

Abstract

In this paper, we consider the Caffarelli–Kohn–Nirenberg (CKN) inequality:

$$\begin{aligned} \bigg (\int _{{\mathbb {R}}^N}|x|^{-b(p+1)}|u|^{p+1}dx\bigg )^{\frac{2}{p+1}}\le C_{a,b,N}\int _{{\mathbb {R}}^N}|x|^{-2a}|\nabla u|^2dx \end{aligned}$$

where $N\ge 3$, $a<\frac{N-2}{2}$, $a\le b\le a+1$ 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) that is positive and radially symmetric in the parameter region $b_{FS}(a)\le b<a+1$ with $a<0$ and $a\le b<a+1$ with $a\ge 0$ 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:

• (1)    stability of CKN inequality in the functional inequality setting

  $$\begin{aligned} dist_{D^{1,2}_{a}}^2(u, {\mathcal {Z}})\lesssim \Vert u\Vert ^2_{D^{1,2}_a({\mathbb {R}}^N)}-C_{a,b,N}^{-1}\Vert u\Vert ^2_{L^{p+1}(|x|^{-b(p+1)},{\mathbb {R}}^N)} \end{aligned}$$

  where ${\mathcal {Z}}= \{ c W_\tau \mid c\in {\mathbb {R}}\backslash \{0\}, \tau >0\}$;

• (2) stability of CKN inequality in the critical point setting (in the class of nonnegative functions)

  $$\begin{aligned} dist_{D_a^{1,2}}(u, {\mathcal {Z}}_0^\nu )\lesssim \left\{ \begin{array}{ll} \Gamma (u),&{} p>2\text { or }\nu =1,\\ \Gamma (u)|\log \Gamma (u)|^{\frac{1}{2}},&{} p=2\text { and }\nu \ge 2,\\ \Gamma (u)^{\frac{p}{2}},&{} 1<p<2\text { and }\nu \ge 2, \end{array}\right. \end{aligned}$$

  where $\Gamma (u)=\Vert div(|x|^{-a}\nabla u)+|x|^{-b(p+1)}|u|^{p-1}u\Vert _{H^{-1}}$ and

  $$\begin{aligned} {\mathcal {Z}}_0^\nu =\{(W_{\tau _1},W_{\tau _2},\ldots ,W_{\tau _\nu })\mid \tau _i>0\}. \end{aligned}$$

Our results generalize the recent work in [7, 11, 15] on the sharp stability of profile decompositions for the special case $a=b=0$ (the Sobolev inequality) to the above parameter region of the Caffarelli–Kohn–Nirenberg inequality. This parameter region is optimal for such stabilities in the sense that in the region $a<b<b_{FS}(a)$ with $a<0$, the nonnegative solution of the Euler–Lagrange equation of CKN inequality is no longer unique.