Title: Sharp stability for critical points of the Sobolev inequality in the absence of bubbling Show Chinese title

Title: 临界点的Sobolev不等式的尖锐稳定性在无爆破解的情况下

Abstract: When $u$ is close to a single Talenti bubble $v$ of the $p$-Sobolev inequality, we show that \begin{equation*} \|Du-Dv\|_{L^p(\mathbb{R}^n)}^{\max\{1,p-1\}}\le C \|-{\rm div}(|Du|^{p-2}Du)-|u|^{p^*-2}u\|_{W^{-1,q}(\mathbb{R}^n)}, \end{equation*} where $C=C(n,p)>0$. This estimate provides a sharp stability estimate for the Struwe-type decomposition in the single bubble case, generalizing the result of Ciraolo, Figalli, and Maggi \cite{CFM2018} (focusing on the case $p=2$) to the arbitrary $p$. Also, in the Sobolev setting, this answers an open problem raised by Zhou and Zou in \cite[Remark 1.17]{ZZ2023}. Show Chinese abstract

Abstract: 当 $u$ 接近 $p$-Sobolev 不等式的单个 Talenti 泡沫 $v$ 时，我们证明了 \begin{equation*} \|Du-Dv\|_{L^p(\mathbb{R}^n)}^{\max\{1,p-1\}}\le C \|-{\rm div}(|Du|^{p-2}Du)-|u|^{p^*-2}u\|_{W^{-1,q}(\mathbb{R}^n)}, \end{equation*} 其中 $C=C(n,p)>0$。 这个估计为单泡情况下的Struwe型分解提供了精确的稳定性估计，推广了Ciraolo、Figalli和Maggi在\cite{CFM2018}中的结果（专注于$p=2$的情况）到任意的$p$。此外，在Sobolev框架下，这回答了Zhou和Zou在\cite[Remark 1.17]{ZZ2023}中提出的一个开放问题。