{
  "id": "2002.06700",
  "version": "v1",
  "published": "2020-02-16T22:56:55.000Z",
  "updated": "2020-02-16T22:56:55.000Z",
  "title": "Non-existence of dead cores in fully nonlinear elliptic models",
  "authors": [
    "Joao Vitor da Silva",
    "Disson dos Prazeres",
    "Humberto Ramos Quoirin"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate non-existence of nonnegative dead-core solutions for the problem $$|Du|^\\gamma F(x, D^2u)+a(x)u^q = 0 \\quad \\mbox{in} \\quad \\Omega, \\quad u=0 \\quad \\mbox{ on } \\quad \\partial\\Omega.$$ Here $\\Omega \\subset \\mathbb{R}^N$ is a bounded smooth domain, $F$ is a fully nonlinear elliptic operator, $a: \\Omega \\to \\mathbb{R}$ is a sign-changing weight, $\\gamma \\geq 0$, and $0<q<\\gamma+1$. We show that this problem has no non-trivial dead core solutions if either $q$ is close enough to $\\gamma+1$ or the negative part of $a$ is sufficiently small. In addition, we obtain the existence and uniqueness of a positive solution under these conditions on $q$ and $a$. Our results extend previous ones established in the semilinear case, and are new even for the simple model $|D u(x)|^{\\gamma} \\mathrm{Tr}(\\mathrm{A}(x) D^2 u(x)) + a(x)u^{q}(x) = 0$, where $\\mathrm{A} \\in C^0(\\Omega;Sym(N))$ is a uniformly elliptic and non-negative matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-16T22:56:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35D40",
      "35J15",
      "35J66",
      "35J70"
    ],
    "keywords": [
      "fully nonlinear elliptic models",
      "non-existence",
      "non-trivial dead core solutions",
      "fully nonlinear elliptic operator",
      "simple model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}