{
  "id": "2005.06513",
  "version": "v1",
  "published": "2020-05-12T12:24:57.000Z",
  "updated": "2020-05-12T12:24:57.000Z",
  "title": "Anisotropic Moser-Trudinger inequality involving $L^{n}$ norm in the entire space $\\mathbb{R}^{n}$",
  "authors": [
    "Rulong Xie"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1703.00901, arXiv:1904.10531 by other authors",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $F: \\mathbb{R}^{n}\\rightarrow [0,+\\infty) $ be a convex function of class $C^{2}( \\mathbb{R}^{n}\\backslash\\{0\\})$ which is even and positively homogeneous of degree 1, and its polar $F^{0}$ represents a Finsler metric on $\\mathbb{R}^{n}$. The anisotropic Sobolev norm in $W^{1,n}\\left(\\mathbb{R}^{n}\\right)$ is defined by \\begin{equation*} ||u||_{F}=\\left(\\int_{\\mathbb{R}^{n}}F^{n}(\\nabla u)+|u|^{n}\\right)^{\\frac{1}{n}}. \\end{equation*} In this paper, the following sharp anisotropic Moser-Trudinger inequality involving $L^{n}$ norm \\[ \\underset{u\\in W^{1,n}( \\mathbb{R}^{n}),\\left\\Vert u\\right\\Vert _{F}\\leq 1}{\\sup}\\int_{ \\mathbb{R} ^{n}}\\Phi\\left( \\lambda_{n}\\left\\vert u\\right\\vert ^{\\frac{n}{n-1}}\\left( 1+\\alpha\\left\\Vert u\\right\\Vert _{n}^{n}\\right) ^{\\frac{1}{n-1}}\\right) dx<+\\infty \\] in the entire space $\\mathbb{R}^n$ for any $0\\leq\\alpha<1$ is established, where $\\Phi\\left( t\\right) =e^{t}-\\underset{j=0}{\\overset{n-2}{\\sum}}% \\frac{t^{j}}{j!}$, $\\lambda_{n}=n^{\\frac{n}{n-1}}\\kappa_{n}^{\\frac{1}{n-1}}$ and $\\kappa_{n}$ is the volume of the unit Wulff ball in $\\mathbb{R}^n$. It is also shown that the above supremum is infinity for all $\\alpha\\geq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $\\alpha>0$ is sufficiently small. The proof of main results in this paper is based on the method of blow-up analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-12T12:24:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "entire space",
      "sharp anisotropic moser-trudinger inequality",
      "unit wulff ball",
      "anisotropic sobolev norm",
      "blow-up analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}