{
  "id": "1710.01271",
  "version": "v1",
  "published": "2017-10-03T16:43:16.000Z",
  "updated": "2017-10-03T16:43:16.000Z",
  "title": "Mass and Extremals Associated with the Hardy-Schrödinger Operator on Hyperbolic Space",
  "authors": [
    "Hardy Chan",
    "Nassif Ghoussoub",
    "Saikat Mazumdar",
    "Shaya Shakerian",
    "Luiz Fernando de Oliveira Faria"
  ],
  "comment": "22 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Hardy-Schr\\\"odinger operator $ -\\Delta_{\\mathbb{B}^n}-\\gamma{V_2}$ on the Poincar\\'e ball model of the Hyperbolic space ${\\mathbb{B}^n}$ ($n \\geq 3$). Here $V_2$ is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at $0$, i.e., $V_2(r)\\sim \\frac{1}{r^2}$. Just like in the Euclidean setting, the operator $ -\\Delta_{\\mathbb{B}^n}-\\gamma{V_2}$ is positive definite whenever $\\gamma <\\frac{(n-2)^2}{4}$, in which case we exhibit explicit solutions for the equation $$-\\Delta_{\\mathbb{B}^n}u-\\gamma{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\\quad{\\text{ in }}\\mathbb{B}^n,$$ where $0\\leq s <2$, $2^*(s)=\\frac{2(n-s)}{n-2}$, and $V_{2^*(s)}$ is a weight that behaves like $\\frac{1}{r^s}$ around $0$. The same equation, on bounded domains $\\Omega$ of ${\\mathbb{B}^n}$ containing $0$ but not touching the hyperbolic boundary, has positive solutions if $0 < \\gamma \\leq \\frac{(n-2)^{2}}{4}-\\frac{1}{4}$. However, if $\\frac{(n-2)^{2}}{4}-\\frac{1}{4}< \\gamma < \\frac{(n-2)^{2}}{4}$, the existence of solutions requires the positivity of the \"hyperbolic Hardy mass\" $m_{_{\\mathbb{B}^n}}(\\Omega)$ of the domain, a notion that we introduce and analyse therein.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-03T16:43:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic space",
      "hardy-schrödinger operator",
      "poincare ball model",
      "hyperbolic hardy mass",
      "chosen radially symmetric potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}