{
  "id": "2008.04455",
  "version": "v1",
  "published": "2020-08-10T23:52:50.000Z",
  "updated": "2020-08-10T23:52:50.000Z",
  "title": "Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations",
  "authors": [
    "Mostafa Fazly",
    "Yuan Li"
  ],
  "comment": "20 pages. Comments welcome",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the quasilinear elliptic equation \\begin{equation*} -Qu=e^u \\ \\ \\text{in} \\ \\ \\Omega\\subset \\mathbb{R}^{N} \\end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by $$Qu:=\\sum_{i=1}^{N}\\frac{\\partial}{\\partial x_{i}}(F(\\nabla u)F_{\\xi_{i}}(\\nabla u)),$$ where $F_{\\xi_{i}}=\\frac{\\partial F}{\\partial\\xi_{i}}$ and $F: \\mathbb{R}^{N}\\rightarrow[0,+\\infty)$ is a convex function of $ C^{2}(\\mathbb{R}^{N}\\setminus\\{0\\})$, that satisfies certain assumptions. For bounded domain $\\Omega$ and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed $N-10$. For the entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions $N<10$ and $2<N<10$, respectively. We also provide an explicit solution that is stable outside a compact set in $N=2$. In addition, we provide similar Liouville theorems for the power-type nonlinearities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-10T23:52:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "anisotropic elliptic equations",
      "stable solutions",
      "partial regularity",
      "finite morse index solutions",
      "quasilinear elliptic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}