{
  "id": "1406.7406",
  "version": "v2",
  "published": "2014-06-28T14:23:51.000Z",
  "updated": "2014-10-09T20:08:48.000Z",
  "title": "Fractional semilinear Neumann problems arising from a fractional Keller--Segel model",
  "authors": [
    "P. R. Stinga",
    "B. Volzone"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We consider the following fractional semilinear Neumann problem on a connected smooth bounded domain $\\Omega\\subset\\mathbb{R}^n$, $n\\geq2$, $$\\begin{cases} (-\\varepsilon\\Delta)^{1/2}u+u=u^p,&\\hbox{in}~\\Omega,\\\\ \\partial_\\nu u=0,&\\hbox{on}~\\partial\\Omega,\\\\ u>0,&\\hbox{in}~\\Omega, \\end{cases}$$ where $\\varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of the semilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. The problem arises by considering steady states of the Keller--Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variational techniques, we prove existence of nonconstant smooth solutions, which are obtained by minimizing a suitable energy functional for small $\\varepsilon$. In the case of large $\\varepsilon$ we obtain nonexistence of nontrivial solutions. It is also shown that as $\\varepsilon\\to0$ the solutions $u_\\varepsilon$ tend to zero in measure on $\\Omega$, while they form spikes in $\\overline{\\Omega}$. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-28T14:23:51.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-09T20:08:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional semilinear neumann problems arising",
      "fractional keller-segel model",
      "nonlocal chemical concentration diffusion",
      "nonconstant smooth solutions",
      "fractional neumann laplacian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.7406S"
    }
  }
}