Existence of an extremal function of Sobolev critical embedding with an $\alpha$-homogeneous weight
Abstract
In our previous publication [{\em Calc. Var. Partial Differential Equations}, 60(1):Paper No. 16, 27, 2021], we delved into examining a critical Sobolev-type embedding of a Sobolev weighted space into an exponential weighted Orlicz space. We specifically determined the optimal Moser-type constant for this embedding, utilizing the monomial weight introduced by Cabr\'e and Ros-Oton [{\em J. Differential Equations}, 255(11):4312--4336, 2013]. Towards the conclusion of that paper, we pledged to explore the existence of an extremal function within this framework. In this current work, we not only provide a positive affirmation to this inquiry but extend it to a broader range of weights known as \emph{-homogeneous weights}.
Keywords
Cite
@article{arxiv.2409.11193,
title = {Existence of an extremal function of Sobolev critical embedding with an $\alpha$-homogeneous weight},
author = {Petr Gurka and Daniel Hauer},
journal= {arXiv preprint arXiv:2409.11193},
year = {2024}
}
Comments
Keywords: Trudinger-Moser inequality, Moser constant, critical Sobolev emebdding, Orlicz exponential space, $\alpha$-homogeneous weight, monomial weight, extremal function