{
  "id": "2409.11193",
  "version": "v1",
  "published": "2024-09-17T13:48:55.000Z",
  "updated": "2024-09-17T13:48:55.000Z",
  "title": "Existence of an extremal function of Sobolev critical embedding with an $α$-homogeneous weight",
  "authors": [
    "Petr Gurka",
    "Daniel Hauer"
  ],
  "comment": "Keywords: Trudinger-Moser inequality, Moser constant, critical Sobolev emebdding, Orlicz exponential space, $\\alpha$-homogeneous weight, monomial weight, extremal function",
  "categories": [
    "math.AP"
  ],
  "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}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-17T13:48:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "46E30",
      "35A23",
      "26D10"
    ],
    "keywords": [
      "extremal function",
      "sobolev critical embedding",
      "homogeneous weight",
      "partial differential equations",
      "exponential weighted orlicz space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}