{
  "id": "1804.04287",
  "version": "v1",
  "published": "2018-04-12T02:21:17.000Z",
  "updated": "2018-04-12T02:21:17.000Z",
  "title": "Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity",
  "authors": [
    "Marius Ghergu",
    "Sunghan Kim",
    "Henrik Shahgholian"
  ],
  "comment": "to appear in Adv. Nonlinear Anal",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the semilinear elliptic equation \\begin{equation*} -\\Delta u=u^\\alpha |\\log u|^\\beta\\quad\\text{in }B_1\\setminus\\{0\\}, \\end{equation*} where $B_1\\subset\\mathbb{R}^n$ with $n\\geq 3$, $\\frac{n}{n-2} < \\alpha < \\frac{n+2}{n-2}$ and $-\\infty<\\beta<\\infty$. Our main result establishes that nonnegative solution $u\\in C^2(B_1\\setminus\\{0\\})$ of the above equation either has a removable singularity at the origin or behaves like \\begin{equation*} u(x) = A(1+o(1)) |x|^{-\\frac{2}{\\alpha-1}} \\left(\\log \\frac{1}{|x|}\\right)^{-\\frac{\\beta}{\\alpha-1}}\\quad\\text{as } x\\rightarrow 0, \\end{equation*} with \\begin{equation*} A=\\left[\\left(\\frac{2}{\\alpha-1}\\right)^{1-\\beta}\\left(n-2-\\frac{2}{\\alpha-1}\\right)\\right]^{\\frac{1}{\\alpha-1}}. \\end{equation*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-12T02:21:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semilinear elliptic equation",
      "exact behavior",
      "log-type nonlinearity",
      "isolated singularity",
      "main result establishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}