Sharp stability for critical points of the Sobolev inequality in the absence of bubbling
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916650187489280 |
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| author | Liu, Gemei Zhang, Yi Ru-Ya |
| author_facet | Liu, Gemei Zhang, Yi Ru-Ya |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp stability for critical points of the Sobolev inequality in the absence of bubbling Liu, Gemei Zhang, Yi Ru-Ya Analysis of PDEs 49J40, 35R20 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}. |
| title | Sharp stability for critical points of the Sobolev inequality in the absence of bubbling |
| topic | Analysis of PDEs 49J40, 35R20 |
| url | https://arxiv.org/abs/2503.02340 |