{
  "id": "2004.06221",
  "version": "v1",
  "published": "2020-04-13T22:14:47.000Z",
  "updated": "2020-04-13T22:14:47.000Z",
  "title": "Some elliptic problems involving the gradient on general bounded and exterior domains",
  "authors": [
    "A. Aghajani",
    "C. Cowan"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \\begin{equation} \\label{eq_abst_1}-\\Delta u = (1+g(x)) | \\nabla u|^p \\qquad \\mbox{ in } B_1, \\qquad u = 0 \\mbox{ on } \\;\\; \\partial B_1, \\qquad \\mbox{ and} \\end{equation} \\begin{equation} \\label{eq_abst_2} -\\Delta u = | \\nabla u|^p \\qquad \\mbox{ in } \\Omega, \\qquad u = 0 \\mbox{ on } \\;\\; \\partial \\Omega. \\end{equation} In the first problem $B_1$ is the unit ball in $ \\mathbb{R}^N$ and in the second $\\Omega$ is a bounded smooth domain in $ \\mathbb{R}^N$. In both cases we assume $ N \\ge 3$, $ \\frac{N}{N-1}<p<2$ and in the first problem we assume $ g \\ge 0$ is a H\\\"older continuous function with $g(0)=0$. We obtain positive singular solutions in both cases. \\\\ For the second equation we also consider the case of $\\Omega$ an exterior domain $ \\mathbb{R}^N$ where $N \\ge 3$ and $ p >\\frac{N}{N-1}$. We prove the existence of a bounded positive classical solution with the additional property that $ \\nabla u(x) \\cdot x>0$ for large $|x|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-13T22:14:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exterior domain",
      "positive singular solutions",
      "elliptic problems",
      "first problem",
      "classical solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}