{
  "id": "1309.5407",
  "version": "v1",
  "published": "2013-09-20T22:45:24.000Z",
  "updated": "2013-09-20T22:45:24.000Z",
  "title": "Nonexistence results for nonlocal equations with critical and supercritical nonlinearities",
  "authors": [
    "Xavier Ros-Oton",
    "Joaquim Serra"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form \\[Lu(x)=\\sum a_{ij}\\partial_{ij}u+{\\rm PV}\\int_{\\R^n}(u(x)-u(x+y))K(y)dy.\\] These operators are infinitesimal generators of symmetric L\\'evy processes. Our results apply to even kernels $K$ satisfying that $K(y)|y|^{n+\\sigma}$ is nondecreasing along rays from the origin, for some $\\sigma\\in(0,2)$ in case $a_{ij}\\equiv0$ and for $\\sigma=2$ in case that $(a_{ij})$ is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for $L$ in star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation to $n$ and $\\sigma$). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian $(-\\Delta)^s$ (here $s>1$) or the fractional $p$-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-20T22:45:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "supercritical nonlinearities",
      "nonlocal equations",
      "higher order fractional laplacian",
      "nonexistence results concern dirichlet problems",
      "nonlocal operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.5407R"
    }
  }
}