Calculus of Variations and Geometric Measure Theory

L. Brasco - E. Lindgren

Uniqueness of extremals for some sharp Poincaré-Sobolev constants

created by brasco on 10 Jan 2022
modified on 18 Aug 2022

[BibTeX]

Submitted Paper

Inserted: 10 jan 2022
Last Updated: 18 aug 2022

Pages: 39
Year: 2022
Notes:

To Peter Lindqvist, a gentleman and $p-$Laplacian master, on the occasion of his 70th birthday.


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 $p$, extremals 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.

Keywords: Nonlinear eigenvalue problems, $p-$Laplacian, Lane-Emden equation, weigthed Sobolev spaces, Poincaré-Sobolev constants


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