{
  "id": "1808.05316",
  "version": "v1",
  "published": "2018-08-16T00:46:09.000Z",
  "updated": "2018-08-16T00:46:09.000Z",
  "title": "A Trudinger-Moser inequality for conical metric in the unit ball",
  "authors": [
    "Yunyan Yang",
    "Xiaobao Zhu"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $\\mathbb{B}$ be the unit ball in $\\mathbb{R}^N$ $(N\\geq 2)$, $p>1$, $g=|x|^{\\frac{2p}{N}\\beta}(dx_1^2+\\cdots+dx_N^2)$ be a conical metric on $\\mathbb{B}$, and $\\lambda_p(\\mathbb{B})=\\inf\\left\\{\\int_\\mathbb{B}|\\nabla u|^Ndx: u\\in W_0^{1,N}(\\mathbb{B}),\\,\\int_\\mathbb{B}|u|^pdx=1\\right\\}$. We prove that for any $\\beta\\geq 0$ and $\\alpha<(1+\\frac{p}{N}\\beta)^{N-1+\\frac{N}{p}}\\lambda_p(\\mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $u\\in W_0^{1,N}(\\mathbb{B})$ with $\\int_\\mathbb{B}|\\nabla u|^Ndx-\\alpha(\\int_\\mathbb{B}|u|^p|x|^{p\\beta}dx)^{N/p}\\leq 1$, there holds $$\\int_\\mathbb{B}e^{\\alpha_N(1+\\frac{p}{N}\\beta)|u|^{\\frac{N}{N-1}}}|x|^{p\\beta}dx\\leq C,$$ where $|x|^{p\\beta}dx=dv_g$, $\\alpha_N=N\\omega_{N-1}^{1/(N-1)}$, $\\omega_{N-1}$ is the area of the unit sphere in $\\mathbb{R}^N$; moreover, extremal functions for such inequalities exist. The case $p=N$, $-1<\\beta<0$ and $\\alpha=0$ was considered by Adimurthi-Sandeep \\cite{A-S}, while the case $p=N=2$, $\\beta\\geq 0$ and $\\alpha=0$ was studied by de Figueiredo-do \\'O-dos Santos \\cite{F-do-dos}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-16T00:46:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "46E35"
    ],
    "keywords": [
      "unit ball",
      "conical metric",
      "trudinger-moser inequality",
      "figueiredo-do o-dos santos",
      "radially symmetric functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}