{
  "id": "1705.05864",
  "version": "v1",
  "published": "2017-05-16T18:01:10.000Z",
  "updated": "2017-05-16T18:01:10.000Z",
  "title": "Extremal functions for the sharp Moser--Trudinger type inequalities in whole space $\\mathbb R^N$",
  "authors": [
    "Van Hoang Nguyen"
  ],
  "comment": "27 pages",
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "This paper is devoted to study the sharp Moser-Trudinger type inequalities in whole space $\\mathbb R^N$, $N \\geq 2$ in more general case. We first compute explicitly the \\emph{normalized vanishing limit} and the \\emph{normalized concentrating limit} of the Moser-Trudinger type functional associated with our inequalities over all the \\emph{normalized vanishing sequences} and the \\emph{normalized concentrating sequences}, respectively. Exploiting these limits together with the concentration-compactness principle of Lions type, we give a proof of the exitence of maximizers for these Moser-Trudinger type inequalities. Our approach gives an alternative proof of the existence of maximizers for the Moser-Trudinger inequality and singular Moser-Trudinger inequality in whole space $\\mathbb R^N$ due to Li and Ruf \\cite{LiRuf2008} and Li and Yang \\cite{LiYang}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-16T18:01:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "26D10"
    ],
    "keywords": [
      "sharp moser-trudinger type inequalities",
      "extremal functions",
      "singular moser-trudinger inequality",
      "moser-trudinger type functional",
      "maximizers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}