{
  "id": "2101.00970",
  "version": "v1",
  "published": "2021-01-04T13:46:04.000Z",
  "updated": "2021-01-04T13:46:04.000Z",
  "title": "Extremal solution and Liouville theorem for anisotropic elliptic equations",
  "authors": [
    "Yuan Li"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the quasilinear Dirichlet boundary problem \\begin{equation}\\nonumber \\left\\{ \\begin{aligned} -Qu&=\\lambda e^{u} \\quad \\mbox{in}\\quad\\Omega\\\\ u&=0 \\quad \\mbox{on}\\quad\\partial\\Omega,\\\\ \\end{aligned} \\right. \\end{equation} where $\\lambda>0$ is a parameter, $\\Omega\\subset\\mathbb{R}^{N}$ with $N\\geq2$ be a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by $$Qu:=\\sum_{i=1}^{N}\\frac{\\partial}{\\partial x_{i}}(F(\\nabla u)F_{\\xi_{i}}(\\nabla u)). $$ Here, $F_{\\xi_{i}}=\\frac{\\partial F}{\\partial\\xi_{i}}$ and $F: \\mathbb{R}^{N}\\rightarrow[0,+\\infty)$ is a convex function of $ C^{2}(\\mathbb{R}^{N}\\setminus\\{0\\})$, that satisfies certain assumptions. We derive the existence of extremal solution and obtain that it's regular, if $N\\leq9$. We also concern the H\\'{e}non type anisotropic Liouville equation, namely, $$-Qu=(F^{0}(x))^{\\alpha}e^{u}\\quad\\mbox{in}\\quad\\mathbb{R}^{N}$$ where $\\alpha>-2$, $N\\geq2$ and $F^{0}$ is the support function of $K:=\\{x\\in\\mathbb{R}^{N}:F(x)<1\\}$ which is defined by $$F^{0}(x):=\\sup_{\\xi\\in K}\\langle x,\\xi\\rangle.$$ We obtain the Liouville theorem for stable solutions and the finite Morse index solutions for $2\\leq N<10+4\\alpha$ and $3\\leq N<10+4\\alpha^{-}$ respectively, where $\\alpha^{-}=\\min\\{\\alpha,0\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-04T13:46:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "anisotropic elliptic equations",
      "liouville theorem",
      "extremal solution",
      "finite morse index solutions",
      "quasilinear dirichlet boundary problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}