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Existence of minimizers of the stability constant of Caffarelli–Kohn–Nirenberg inequality

Part of: Existence theories Spectral theory and eigenvalue problems Inequalities

Published online by Cambridge University Press:  09 September 2025

Shengbing Deng ,
Xingliang Tian  and
Juncheng Wei
Shengbing Deng*
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China (shbdeng@swu.edu.cn)
Xingliang Tian
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China (xltian@email.swu.edu.cn)
Juncheng Wei
Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories 999077, Hong Kong (wei@math.cuhk.edu.hk)
*
*Corresponding author.
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Abstract

In this paper, we prove the existence of minimizers for the sharp stability constant of Caffarelli–Kohn–Nirenberg inequality near the new curve $b^*_{\mathrm{FS}}(a)$ (which lies above the well-known Felli–Schneider curve $b_{\mathrm{FS}}(a)$), extending the work of Wei and Wu [Math. Z., 2024] to a slightly larger region. Moreover, we provide an upper bound for the Caffarelli–Kohn–Nirenberg inequality with an explicit sharp constant, which may have its own interest.

Keywords

Caffarelli–Kohn–Nirenberg inequalityFelli–Schneider curveexistence of minimizersupper bound of inequalityBianchi–Egnell type stability

MSC classification

Primary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory
Secondary: 26D10: Inequalities involving derivatives and differential and integral operators 49J40: Variational methods including variational inequalities

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 23
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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