Stability of the Caffarelli-Kohn-Nirenberg inequality: the existence of minimizers

Juncheng Wei, Yuanze Wu

Published 2023-08-09Version 1

In this paper, we consider the following variational problem: $\begin{eqnarray*} \inf_{u\in D^{1,2}_a(\bbr^N)\backslash\mathcal{Z}}\frac{\|u\|^2_{D^{1,2}_a(\bbr^N)}-C_{a,b,N}^{-1}\|u\|^2_{L^{p+1}(|x|^{-b(p+1)},\bbr^N)}}{dist_{D^{1,2}_{a}}^2(u, \mathcal{Z})}:=c_{BE}, \end{eqnarray*}$ where $N\geq2$, $b_{FS}(a)<b<a+1$ with $b_{FS}(a)$ being the Felli-Schneider curve, $a<\frac{N-2}{2}$, $p=\frac{N+2(1+a-b)}{N-2(1+a-b)}$, $\mathcal{Z}= \{ c \tau^{\frac{N-2-2a}{2}}W(\tau x)\mid c\in\bbr\backslash\{0\}, \tau>0\}$ and up to dilations and scalar multiplications (also up to translations in the special case $a=b=0$), $W(x)$, which is positive and radially symmetric, is the unique extremal function of the following classical Caffarelli-Kohn-Nirenberg (CKN for short) inequality \begin{eqnarray*} \bigg(\int_{\bbr^N}|x|^{-b(p+1)}|u|^{p+1}dx\bigg)^{\frac{2}{p+1}}\leq C_{a,b,N}\int_{\bbr^N}|x|^{-2a}|\nabla u|^2dx \end{eqnarray*} with $C_{a,b,N}$ being the optimal constant. It is known in \cite{WW2022} that $c_{BE}>0$. In this paper, we prove that the above variational problem has a minimizer for $N\geq3$, which extends the result of Konig in \cite{K2023} for the Sobolev inequality to the CKN inequality. Moreover, we believe that our restriction on dimensions is optimal for the existence of minimizers of the above variational problem and believe that there is no minimizers of the above variational problem for $N=2$ with $c_{BE}=\frac{2(p-1)}{3p-1}$.