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arXiv · 2409.11193

Existence of an extremal function of Sobolev critical embedding with an $\alpha$-homogeneous weight

Petr Gurka·Daniel Hauer
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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{$\alpha$-homogeneous weights}.

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Petr Gurka, Daniel Hauer. 2024-09-17. Existence of an extremal function of Sobolev critical embedding with an $\alpha$-homogeneous weight. https://arxiv.org/abs/2409.11193

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