{
  "id": "2009.00157",
  "version": "v1",
  "published": "2020-09-01T00:41:59.000Z",
  "updated": "2020-09-01T00:41:59.000Z",
  "title": "Sharp existence and classification results for nonlinear elliptic equations in $\\mathbb R^N\\setminus\\{0\\}$ with Hardy potential",
  "authors": [
    "Florica C. Cîrstea",
    "Maria Fărcăşeanu"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "For $N\\geq 3$, by the seminal paper of Brezis and V\\'eron (Arch. Rational Mech. Anal. 75(1):1--6, 1980/81), no positive solutions of $-\\Delta u+u^q=0$ in $\\mathbb R^N\\setminus \\{0\\}$ exist if $q\\geq N/(N-2)$; for $1<q<N/(N-2)$ the existence and profiles near zero of all positive $C^1(\\mathbb R^N\\setminus \\{0\\})$ solutions are given by Friedman and V\\'eron (Arch. Rational Mech. Anal. 96(4):359--387, 1986). In this paper, for every $q>1$ and $\\theta\\in \\mathbb R$, we prove that the nonlinear elliptic problem (*) $-\\Delta u-\\lambda \\,|x|^{-2}\\,u+|x|^{\\theta}u^q=0$ in $\\mathbb R^N\\setminus \\{0\\}$ with $u>0$ has a $C^1(\\mathbb R^N\\setminus \\{0\\})$ solution if and only if $\\lambda>\\lambda^*$, where $\\lambda^*=\\Theta(N-2-\\Theta) $ with $\\Theta=(\\theta+2)/(q-1)$. We show that (a) if $\\lambda>(N-2)^2/4$, then $U_0(x)=(\\lambda-\\lambda^*)^{1/(q-1)}|x|^{-\\Theta}$ is the only solution of (*) and (b) if $\\lambda^*<\\lambda\\leq (N-2)^2/4$, then all solutions of (*) are radially symmetric and their total set is $U_0\\cup \\{U_{\\gamma,q,\\lambda}:\\ \\gamma\\in (0,\\infty) \\}$. We give the precise behavior of $ U_{\\gamma,q,\\lambda}$ near zero and at infinity, distinguishing between $1<q<q_{N,\\theta}$ and $q>\\max\\{q_{N,\\theta},1\\}$, where $q_{N,\\theta}=(N+2\\theta+2)/(N-2)$. In addition, for $\\theta\\leq -2$ we settle the structure of the set of all positive solutions of (*) in $\\Omega\\setminus \\{0\\}$, subject to $u|_{\\partial\\Omega}=0$, where $\\Omega$ is a smooth bounded domain containing zero, complementing the works of C\\^{\\i}rstea (Mem. Amer. Math. Soc. 227, 2014) and Wei--Du (J. Differential Equations 262(7):3864--3886, 2017).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-01T00:41:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35B40",
      "35J25"
    ],
    "keywords": [
      "nonlinear elliptic equations",
      "sharp existence",
      "classification results",
      "hardy potential",
      "bounded domain containing zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}