On the stability of Caffarelli-Kohn-Nirenberg inequality in $\R^2$

Dolbeault, Esteban and Loss \cite{DEL16} obtained an optimal rigidity result, that is, when $a<0$ and $b_{\mathrm{FS}}(a)\leq b<a+1$ the extremal function for best constant $\mathcal{S}_{a,b}>0$ of the following Caffarelli-Kohn-Nirenberg inequality is symmetry, \[ \mathcal{S}_{a,b}\left(\int_{\R^2}|x|^{-qb}|u|^q \mathrm{d}x\right)^{\frac{2}{q}} \leq \int_{\R^2}|x|^{-2a}|\nabla u|^2 \mathrm{d}x, \quad \mbox{for all}\quad u\in C^\infty_0(\R^2), \] where $b_{\mathrm{FS}}(a):=a-\frac{a}{\sqrt{a^2+1}}$, $q=\frac{2}{b-a}$. An important task is investigating the stability of critical points set $\mathcal{M}$ for this inequality. Firstly, we classify solutions of the linearized problem related to the extremals which fills the work of Felli and Schneider \cite{FS03}. When $b_{\mathrm{FS}}(a)<b<a+1$, we investigate the stability of previous inequality by using spectral estimate combined with a compactness argument that \begin{align*} \int_{\mathbb{R}^2}|x|^{-2a}|\nabla u|^2 \mathrm{d}x -\mathcal{S}_{a,b}\left(\int_{\mathbb{R}^2}|x|^{-qb}|u|^q \mathrm{d}x\right)^{\frac{2}{q}} \geq \mathcal{B} \mathrm{dist}(u,\mathcal{M})^2,\quad \mbox{for all}\quad u\in C^\infty_0(\R^2), \end{align*} for some $\mathcal{B}>0$, however it is false when $b=b_{\mathrm{FS}}(a)$, which extends the work of Wei and Wu \cite{WW22} to $\R^2$. Furthermore, we obtain the existence of minimizers for $\mathcal{B}$ which extends the recent work of K\"{o}nig \cite{Ko22-2}.