{
  "id": "2112.01755",
  "version": "v2",
  "published": "2021-12-03T07:25:24.000Z",
  "updated": "2023-02-15T05:21:28.000Z",
  "title": "Positive solutions of the $\\mathcal{A}$-Laplace equation with a potential",
  "authors": [
    "Yongjun Hou",
    "Yehuda Pinchover",
    "Antti Rasila"
  ],
  "comment": "43 pages",
  "journal": "Potential Analysis (2023)",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study positive solutions of the quasilinear elliptic equation $$ Q'_{p,\\mathcal{A},V}[u]\\triangleq -\\dive{\\mathcal{A}(x,\\nabla u)}+V(x)|u|^{p-2}u=0, $$ in a domain $\\Gw\\subseteq \\R^n$, where $n\\geq 2$, $1<p<\\infty$, the divergence of~$\\mathcal{A}$ is the well known $\\mathcal{A}$-Laplace operator considered in the influential book of Heinonen, Kilpel\\\"{a}inen, and Martio, and the potential $V$ belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator $Q'_{p,\\mathcal{A},V}$. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of $Q'_{p,\\mathcal{A},V}$ in a domain $\\omega\\Subset\\Omega$, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation $Q'_{p,\\mathcal{A},V}[u]=0$ of minimal growth at infinity in $\\Omega$, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-02-15T05:21:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B09",
      "35B50",
      "35J08",
      "35J62"
    ],
    "keywords": [
      "laplace equation",
      "study positive solutions",
      "minimal positive green function",
      "local morrey space",
      "quasilinear elliptic equation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}