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

TL;DR

This paper proves the existence of minimizers for a variational problem related to the Caffarelli-Kohn-Nirenberg inequality, extending previous results and identifying optimal conditions for minimizer existence.

Contribution

It establishes the existence of minimizers under specific parameter conditions, extending prior work on Sobolev inequalities to the CKN inequality.

Findings

Minimizers exist under certain parameter ranges.

Conditions for existence are shown to be optimal.

Results extend previous Sobolev inequality findings.

Abstract

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 $N\geq2$, $b_{FS}(a)<b<a+1$ for $a<0$ and $a\leq b<a+1$ for $0\leq a<a_c:=\frac{N-2}{2}$ and $a+b>0$ with $b_{FS}(a)$ being the Felli-Schneider curve, $p=\frac{N+2(1+a-b)}{N-2(1+a-b)}$, $\mathcal{Z}= \{ c \tau^{a_c-a}W(\tau x)\mid c\in\bbr\backslash\{0\}, \tau>0\}$ and up to dilations and scalar multiplications, $W(x)$, 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 $N\geq2$ under the following two assumptions: \begin{enumerate} \item[$(i)$]\quad $a_c^*\leq a<a_c$ and $a\leq b<a+1$, \item[$(ii)$]\quad $a<a_c^*$ and $b_{FS}^*(a)\leq b<a+1$, \end{enumerate} where $a_c^*=\bigg(1-\sqrt{\frac{N-1}{2N}}\bigg)a_c$ and \begin{eqnarray*} b_{FS}^*(a)=\frac{(a_c-a)N}{a_c-a+\sqrt{(a_c-a)^2+N-1}}+a-a_c. \end{eqnarray*} Our results extend that of Konig in \cite{K2023} for the Sobolev inequality to the CKN inequality. Moreover, we believe that our assumptions~$(i)$ and $(ii)$ are optimal for the existence of minimizers of the above variational problem.