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 17 Oct 2022


Accepted Paper

Inserted: 10 jan 2022
Last Updated: 17 oct 2022

Journal: Trans. Amer. Math. Soc.
Pages: 39
Year: 2022

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


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