{
  "id": "2002.12170",
  "version": "v1",
  "published": "2020-02-26T09:10:53.000Z",
  "updated": "2020-02-26T09:10:53.000Z",
  "title": "Classification of radial solutions for elliptic systems driven by the $k$-Hessian operator",
  "authors": [
    "Marius Ghergu"
  ],
  "comment": "21 pages. arXiv admin note: text overlap with arXiv:1808.00407",
  "categories": [
    "math.AP"
  ],
  "abstract": "We are concerned with non-constant positive radial solutions of the system $$ \\left\\{ \\begin{aligned} S_k(D^2 u)&=|\\nabla u|^{m} v^{p}&&\\quad\\mbox{ in }\\Omega,\\\\ S_k(D^2 v)&=|\\nabla u|^{q} v^{s} &&\\quad\\mbox{ in }\\Omega, \\end{aligned} \\right. $$ where $S_k(D^2u)$ is the $k$-Hessian operator of $u\\in C^2(\\Omega)$ ($1\\leq k\\leq N$) and $\\Omega\\subset\\mathbb{R}^N$ $(N\\geq 2)$ is either a ball or the whole space. The exponents satisfy $q>0$, $m,s\\geq 0$, $p\\geq s\\geq 0$ and $(k-m)(k-s)\\neq pq$. In the case where $\\Omega$ is a ball, we classify all the positive radial solutions according to their behavior at the boundary. Further, we consider the case $\\Omega=\\mathbb{R}^N$ and find that the above system admits non-constant positive radial solutions if and only if $0\\leq m<k$ and $pq < (k-m)(k-s)$. Using arguments from three component cooperative and irreducible dynamical systems we deduce the behavior at infinity of such solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-26T09:10:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J47",
      "35B40",
      "70G60"
    ],
    "keywords": [
      "elliptic systems driven",
      "hessian operator",
      "classification",
      "system admits non-constant positive radial",
      "admits non-constant positive radial solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}