{
  "id": "1505.02490",
  "version": "v1",
  "published": "2015-05-11T05:44:04.000Z",
  "updated": "2015-05-11T05:44:04.000Z",
  "title": "Boundary blow-up solutions to fractional elliptic equations in a measure framework",
  "authors": [
    "Huyuan Chen",
    "Hichem Hajaiej",
    "Ying Wang"
  ],
  "comment": "25 pages. arXiv admin note: text overlap with arXiv:1410.2672",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\alpha\\in(0,1)$, $\\Omega$ be a bounded open domain in $R^N$ ($N\\ge 2$) with $C^2$ boundary $\\partial\\Omega$ and $\\omega$ be the Hausdorff measure on $\\partial\\Omega$. We denote by $\\frac{\\partial^\\alpha \\omega}{\\partial \\vec{n}^\\alpha}$ a measure $$\\langle\\frac{\\partial^\\alpha \\omega}{\\partial \\vec{n}^\\alpha},f\\rangle=\\int_{\\partial\\Omega}\\frac{\\partial^\\alpha f(x)}{\\partial \\vec{n}_x^\\alpha} d\\omega(x),\\quad f\\in C^1(\\bar\\Omega),$$ where $\\vec{n}_x$ is the unit outward normal vector at point $x\\in\\partial\\Omega$. In this paper, we prove that problem $$ \\begin{array}{lll} (-\\Delta)^\\alpha u+g(u)=k\\frac{\\partial^\\alpha \\omega}{\\partial \\vec{n}^\\alpha}\\quad & {\\rm in}\\quad \\bar\\Omega,\\\\[2mm] \\phantom{(-\\Delta)^\\alpha +g(u)} u=0\\quad & {\\rm in}\\quad \\Omega^c \\end{array} $$ admits a unique weak solution $u_k$ under the hypotheses that $k>0$, $(-\\Delta)^\\alpha$ denotes the fractional Laplacian with $\\alpha\\in(0,1)$ and $g$ is a nondecreasing function satisfying extra conditions. We prove that the weak solution is a classical solution of $$ \\begin{array}{lll} \\ \\ \\ (-\\Delta)^\\alpha u+g(u)=0\\quad & {\\rm in}\\quad \\Omega,\\\\[2mm] \\phantom{------\\} \\ u=0\\quad & {\\rm in}\\quad R^N\\setminus\\bar\\Omega,\\\\[2mm] \\phantom{} \\lim_{x\\in\\Omega,x\\to\\partial\\Omega}u(x)=+\\infty. \\end{array} $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-11T05:44:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional elliptic equations",
      "boundary blow-up solutions",
      "measure framework",
      "unit outward normal vector",
      "nondecreasing function satisfying extra conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150502490C"
    }
  }
}