{
  "id": "1804.05991",
  "version": "v1",
  "published": "2018-04-17T00:17:27.000Z",
  "updated": "2018-04-17T00:17:27.000Z",
  "title": "The Hardy--Schrödinger Operator on the Poincaré Ball: Compactness and Multiplicity",
  "authors": [
    "Nassif Ghoussoub",
    "Saikat Mazumdar",
    "Frédéric Robert"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a compact smooth domain in the Poincar\\'e ball model of the Hyperbolic space $\\mathbb{B}^n$, $n \\geq 5$. Let $0< s <2$ and write $2^{\\star}(s):=\\frac{2(n-s)}{n-2}$ for the corresponding critical Sobolev exponent. We show that if $\\gamma<\\frac{(n-2)^2}{4}-4$ and $ \\lambda > \\frac{n-2}{n-4} \\left(\\frac{n(n-4)}{4}-\\gamma \\right)$, then the Dirichlet boundary value problem: \\begin{eqnarray*} \\left\\{ \\begin{array}{lll} -\\Delta_{\\mathbb{B}^n}u-\\gamma{V_2}u -\\lambda u&=V_{2^{\\star}(s)}|u|^{2^{\\star}(s)-2}u &\\hbox{ in }\\Omega \\hfill u &=0 & \\hbox{ on } \\partial \\Omega, \\end{array} \\right. \\end{eqnarray*} has infinitely many solutions. Here $-\\Delta_{\\mathbb{B}^n}$ is the Laplace-Beltrami operator on $\\mathbb{B}^n$, $V_{2}$ is a Hardy-type potential that behaves like $\\frac{1}{r^{2}}$ at the origin, while $V_{2^{\\star}(s)}$ is the Hardy-Sobolev weight, which behaves like $\\frac{1}{r^{s}}$ at the origin. The solutions belong to $ C^{2}(\\overline{\\Omega}\\setminus\\{0\\})$, while around $0$ they behave like \\begin{align*} u(x) \\sim \\frac{c(n,\\gamma)}{|x|^{\\frac{n-2}{2}-\\sqrt{\\frac{(n-2)^{2}}{4}-\\gamma}}}, \\end{align*} where $c(n,\\gamma)$ is some constant in $\\mathbb{R}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-17T00:17:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy-schrödinger operator",
      "compactness",
      "multiplicity",
      "dirichlet boundary value problem",
      "poincare ball model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}