{
  "id": "1111.0796",
  "version": "v1",
  "published": "2011-11-03T11:13:42.000Z",
  "updated": "2011-11-03T11:13:42.000Z",
  "title": "Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions",
  "authors": [
    "Xavier Cabre",
    "Yannick Sire"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper, which is the follow-up to part I, concerns the equation $(-\\Delta)^{s} v+G'(v)=0$ in $\\mathbb{R}^{n}$, with $s \\in (0,1)$, where $(-\\Delta)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a L\\'evy process. When $n=1$, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits $\\pm 1$ at $\\pm \\infty$) if and only if the potential $G$ has only two absolute minima in $[-1,1]$, located at $\\pm 1$ and satisfying $G'(-1)=G'(1)=0$. Under the additional hypothesis $G\"(-1)>0$ and $G\"(1)>0$, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution. For $n\\geq 1$, we prove some results related to the one-dimensional symmetry of certain solutions ---in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-03T11:13:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplacian",
      "nonlinear equations",
      "qualitative properties",
      "uniqueness",
      "layer solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.0796C"
    }
  }
}