Sharp stability for critical points of the Sobolev inequality in the absence of bubbling
Gemei Liu, Yi Ru-Ya Zhang
TL;DR
This work gives a sharp stability estimate for the Sobolev inequality in the absence of bubbling, focusing on single-bubble configurations near Talenti bubbles. The authors develop new vector inequalities and spectral-gap analyses for the nonlinear $p$-Laplacian to control the gradient distance between a near-minimizer $u$ and the corresponding bubble $v$, achieving the optimal exponent $\max\{1,\,p-1\}$ in the bound $\|Du-Dv\|_{L^p}^{\max\{1,p-1\}}\le C\|P(u)\|_{W^{-1,q}}$, where $P(u)=-\operatorname{div}(|Du|^{p-2}Du)-|u|^{p^*-2}u$. Building on past stability results for the Sobolev energy and Struwe's decomposition, the paper extends the theory to arbitrary $1<p<n$, obtaining a sharp, single-bubble stability in the Euler–Lagrange context and addressing an open problem of Zhou and Zou in the Sobolev setting. The approach combines carefully crafted vector inequalities with a refined spectral-gap framework, enabling precise control of the nonlinear terms and the perturbative analysis around a Talenti bubble. The results have implications for quantitative stability in geometric analysis and for understanding nearly optimal solutions in critical Sobolev problems.
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}.