Sharp stability for critical points of the Sobolev inequality in the absence of bubbling

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}.