Uniqueness of extremals for some sharp Poincaré-Sobolev constants
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- by Lorenzo Brasco and Erik Lindgren;
- Trans. Amer. Math. Soc. 376 (2023), 3541-3584
- DOI: https://doi.org/10.1090/tran/8838
- Published electronically: February 9, 2023
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Abstract:
We study the sharp constant for the embedding of $W^{1,p}_0(\Omega )$ into $L^q(\Omega )$, in the case $2<p<q$. We prove that for smooth connected sets, when $q>p$ and $q$ is sufficiently close to $p$, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side.
The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of $p-$Laplace–type equations by L. Damascelli and B. Sciunzi.
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Bibliographic Information
- Lorenzo Brasco
- Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy
- MR Author ID: 884059
- Email: lorenzo.brasco@unife.it
- Erik Lindgren
- Affiliation: Department of Mathematics, KTH – Royal Institute of Technology, 100 44 Stockholm, Sweden
- MR Author ID: 818487
- Email: eriklin@math.kth.se
- Received by editor(s): February 16, 2022
- Received by editor(s) in revised form: August 19, 2022, and October 10, 2022
- Published electronically: February 9, 2023
- Additional Notes: The second author was supported by the Swedish Research Council, grant no. 2017-03736.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 3541-3584
- MSC (2020): Primary 35P30, 35A02, 35B65
- DOI: https://doi.org/10.1090/tran/8838
- MathSciNet review: 4577341
Dedicated: To Peter Lindqvist, a gentleman and $p-$Laplacian master, on the occasion of his 70th birthday