Stability of Caffarelli-Kohn-Nirenberg inequality

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 $N\ge 3$, $a<\frac{N-2}{2}$, $a\le b\le a+1$ 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 $W(x)$ which 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: \begin{enumerate} \item[$(1)$] \quad stability of CKN inequality in the functional inequality setting $$dis{t}_{D_a^{1,2}}^2(u,\mathcal{Z})\lesssim \Vert u{\Vert }_{D_a^{1,2}({\mathbb{R}}^N)}^2-C_{a,b,N}^{-1}\Vert u{\Vert }_{L^{p+1}(|x{|}^{-b(p+1)},{\mathbb{R}}^N)}^2$$ where $\mathcal{Z}= \{ c W_τ\mid c\in\bbr\backslash\{0\}, τ>0\}$; \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^ν)\lesssim\left\{\aligned &Γ(u),\quad p>2\text{ or }ν=1,\\ &Γ(u)|\logΓ(u)|^{\frac12},\quad p=2\text{ and }ν\geq2,\\ &Γ(u)^{\frac{p}{2}},\quad 1<p<2\text{ and }ν\geq2, \endaligned\right. \end{eqnarray*} where $\Gamma (u)=\Vert div(|x{|}^{-a}\nabla u)+|x{|}^{-b(p+1)}|u{|}^{p-1}u{\Vert }_{(D_a^{1,2}{)}^{{}^{\prime }}}$ and $${\mathcal{Z}}_0^{\nu }=\{(W_{{\tau }_1},W_{{\tau }_2},\cdots ,W_{{\tau }_{\nu }})\mid {\tau }_i>0\}.$$

29 pages; comments welcome

References in corpus (1)

• Sharp quantitative estimates of Struwe's Decomposition