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

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arxiv 2503.02340 v2 pith:OL47R4TL submitted 2025-03-04 math.AP

Gemei Liu , Yi Ru-Ya Zhang This is my paper · ORCID
classification math.AP
keywords sobolevbubblecaseciteequationestimateinequalitymathbb
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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}.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality

    math.AP 2026-08 accept novelty 8.0 of 10

    Positive entire solutions of the critical p-Laplace equation with monotone coefficient h are always explicit Talenti bubbles, and a scale-invariant Harnack inequality holds.

  2. Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$

    math.AP 2026-07 accept novelty 7.0 of 10

    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.

  3. Sharp quantitative stability estimates for the Brezis-Nirenberg problem

    math.AP 2025-06 conditional novelty 7.0 of 10

    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.

  4. Sharp One-bubble Critical-Point Stability and Global Compactness for the Sobolev Trace Inequality

    math.AP 2026-07 conditional novelty 5.0 of 10

    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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