{
  "id": "1311.6952",
  "version": "v1",
  "published": "2013-11-27T12:33:30.000Z",
  "updated": "2013-11-27T12:33:30.000Z",
  "title": "Radial symmetry of positive solutions involving the fractional Laplacian",
  "authors": [
    "Patricio Felmer",
    "Ying Wang"
  ],
  "comment": "27 pages, Communications in Contemporary Mathematics 2013",
  "categories": [
    "math.AP"
  ],
  "abstract": "The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem (-\\Delta)^{\\alpha} u=f(u)+g,\\ \\ {\\rm{in}}\\ \\ B_1, \\quad u=0\\ \\ {\\rm in}\\ \\ B_1^c, where $(-\\Delta)^\\alpha$ denotes the fractional Laplacian, $\\alpha\\in(0,1)$, and $B_1$ denotes the open unit ball centered at the origin in $\\R^N$ with $N\\ge2$. The function $f:[0,\\infty)\\to\\R$ is assumed to be locally Lipschitz continuous and $g: B_1\\to\\R$ is radially symmetric and decreasing in $|x|$. In the second place we consider radial symmetry of positive solutions for the equation (-\\Delta)^{\\alpha} u=f(u),\\ \\ {\\rm{in}}\\ \\ \\R^N, with $u$ decaying at infinity and $f$ satisfying some extra hypothesis, but possibly being non-increasing. Our third goal is to consider radial symmetry of positive solutions for system of the form (-\\Delta)^{\\alpha_1} u=f_1(v)+g_1,\\ \\ \\ \\ & {\\rm{in}}\\quad B_1,\\\\[2mm] (-\\Delta)^{\\alpha_2} v=f_2(u)+g_2,\\ \\ \\ \\ & {\\rm{in}} \\quad B_1,\\\\[2mm] u=v =0,\\ \\ \\ \\ & {\\rm{in}}\\quad B_1^c, where $\\alpha_1,\\alpha_2\\in(0,1)$, the functions $f_1$ and $f_2$ are locally Lipschitz continuous and increasing in $[0,\\infty)$, and the functions $g_1$ and $g_2$ are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-27T12:33:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplacian",
      "positive solution",
      "locally lipschitz continuous",
      "study radial symmetry",
      "semi-linear dirichlet problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.6952F"
    }
  }
}