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Sharp stability for critical points of the Sobolev inequality in the absence of bubbling
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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}.
Forward citations
Cited by 3 Pith papers
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Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$
Sharp stability estimates with optimal exponents are established for the affine Sobolev inequality and its critical points for p≥2, including a new affine spectral gap inequality.
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Sharp quantitative stability estimates for the Brezis-Nirenberg problem
Nearly stationary functions for the Brezis-Nirenberg problem on bounded domains lie within a sharp, dimension-dependent distance of a solution plus bubbles, and the optimal exponents are identified.
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Sharp One-bubble Critical-Point Stability and Global Compactness for the Sobolev Trace Inequality
Near one trace bubble, the Euler-Lagrange residual controls the L^p-gradient distance with sharp power max{1,p-1}, and a Struwe-type compactness decomposition holds.
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