{
  "id": "1106.5137",
  "version": "v1",
  "published": "2011-06-25T14:30:37.000Z",
  "updated": "2011-06-25T14:30:37.000Z",
  "title": "On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators",
  "authors": [
    "Jerome Coville"
  ],
  "journal": "J. Differential Equations 249 (2010) 2921-2953",
  "doi": "10.1016/j.jde.2010.07.003",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "In this paper we are interested in the existence of a principal eigenfunction of a nonlocal operator which appears in the description of various phenomena ranging from population dynamics to micro-magnetism. More precisely, we study the following eigenvalue problem: $$\\int_{\\O}J(\\frac{x-y}{g(y)})\\frac{\\phi(y)}{g^n(y)}\\, dy +a(x)\\phi =\\rho \\phi,$$ where $\\O\\subset\\R^n$ is an open connected set, $J$ a nonnegative kernel and $g$ a positive function. First, we establish a criterion for the existence of a principal eigenpair $(\\lambda_p,\\phi_p)$. We also explore the relation between the sign of the largest element of the spectrum with a strong maximum property satisfied by the operator. As an application of these results we construct and characterize the solutions of some nonlinear nonlocal reaction diffusion equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-06-25T14:30:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "principal eigenfunction",
      "nonlocal operator",
      "simple criterion",
      "nonlinear nonlocal reaction diffusion equations",
      "strong maximum property"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.5137C"
    }
  }
}