{"title":"Existence of an extremal function of Sobolev critical embedding with an $α$-homogeneous weight","authors":"Petr Gurka, Daniel Hauer","doi":"arxiv-2409.11193","DOIUrl":null,"url":null,"abstract":"In our previous publication [{\\em Calc. Var. Partial Differential Equations},\n60(1):Paper No. 16, 27, 2021], we delved into examining a critical Sobolev-type\nembedding of a Sobolev weighted space into an exponential weighted Orlicz\nspace. We specifically determined the optimal Moser-type constant for this\nembedding, utilizing the monomial weight introduced by Cabr\\'e and Ros-Oton\n[{\\em J. Differential Equations}, 255(11):4312--4336, 2013]. Towards the\nconclusion of that paper, we pledged to explore the existence of an extremal\nfunction within this framework. In this current work, we not only provide a positive affirmation to this\ninquiry but extend it to a broader range of weights known as\n\\emph{$\\alpha$-homogeneous weights}.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Analysis of PDEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11193","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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}.