{
  "id": "1810.03055",
  "version": "v1",
  "published": "2018-10-06T21:02:18.000Z",
  "updated": "2018-10-06T21:02:18.000Z",
  "title": "Superlinear elliptic inequalities on manifolds",
  "authors": [
    "Alexander Grigor'yan",
    "Yuhua Sun",
    "Igor Verbitsky"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $M$ be a complete non-compact Riemannian manifold and let $\\sigma $ be a Radon measure on $M$. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy \\begin{equation*} -\\Delta u\\geq \\sigma u^{q}\\quad \\text{in}\\,\\,M, \\end{equation*} where $q>1$. We obtain necessary and sufficent criteria for existence of positive solutions in terms of Green function of $\\Delta $. In particular, explicit necessary and sufficient conditions are given when $M$ has nonnegative Ricci curvature everywhere in $M$, or more generally when Green's function satisfies the 3G-inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-06T21:02:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J61",
      "58J05",
      "31B10",
      "42B37"
    ],
    "keywords": [
      "superlinear elliptic inequalities",
      "complete non-compact riemannian manifold",
      "greens function satisfies",
      "positive solutions",
      "semilinear elliptic inequaliy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}