{
  "id": "1901.00412",
  "version": "v1",
  "published": "2019-01-02T15:20:52.000Z",
  "updated": "2019-01-02T15:20:52.000Z",
  "title": "Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains",
  "authors": [
    "Wei Dai",
    "Guolin Qin"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1810.02752",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we are mainly concerned with the Dirichlet problems in exterior domains for the following elliptic equations: \\begin{equation}\\label{GPDE0} (-\\Delta)^{\\frac{\\alpha}{2}}u(x)=f(x,u) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\text{in} \\,\\,\\,\\, \\Omega_{r}:=\\{x\\in\\mathbb{R}^{n}\\,|\\,|x|>r\\} \\end{equation} with arbitrary $r>0$, where $n\\geq2$, $0<\\alpha\\leq 2$ and $f(x,u)$ satisfies some assumptions. A typical case is the Hardy-H\\'{e}non type equations in exterior domains. We first derive the equivalence between \\eqref{GPDE0} and the corresponding integral equations \\begin{equation}\\label{GIE0} u(x)=\\int_{\\Omega_{r}}G_{\\alpha}(x,y)f(y,u(y))dy, \\end{equation} where $G_{\\alpha}(x,y)$ denotes the Green's function for $(-\\Delta)^{\\frac{\\alpha}{2}}$ in $\\Omega_{r}$ with Dirichlet boundary conditions. Then, we establish Liouville theorems for \\eqref{GIE0} via the method of scaling spheres developed in \\cite{DQ0} by Dai and Qin, and hence obtain the Liouville theorems for \\eqref{GPDE0}. Liouville theorems for integral equations related to higher order Navier problems in $\\Omega_{r}$ are also derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-02T15:20:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B53",
      "35J30",
      "35J91"
    ],
    "keywords": [
      "liouville type theorems",
      "exterior domains",
      "elliptic equations",
      "dirichlet conditions",
      "higher order navier problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}