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

TL;DR

This paper proves the existence of extremal functions for a Sobolev critical embedding involving -homogeneous weights, extending previous results on weighted Sobolev spaces and optimal constants.

Contribution

It establishes the existence of extremal functions for Sobolev embeddings with -homogeneous weights, broadening the scope of prior work on weighted Sobolev inequalities.

Findings

Confirmed existence of extremal functions for -homogeneous weights.

Extended previous results on optimal constants in weighted Sobolev embeddings.

Generalized the class of weights for which extremal functions are known to exist.

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}.