{
  "id": "1712.08598",
  "version": "v1",
  "published": "2017-12-22T17:55:36.000Z",
  "updated": "2017-12-22T17:55:36.000Z",
  "title": "Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian",
  "authors": [
    "Tomás Sanz-Perela"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the regularity of stable solutions to the problem $$ \\left\\{ \\begin{array}{rcll} (-\\Delta)^s u &=& f(u) & \\text{in} \\quad B_1\\,, u &\\equiv&0 & \\text{in} \\quad \\mathbb R^n\\setminus B_1\\,, \\end{array} \\right. $$ where $s\\in(0,1)$. Our main result establishes an $L^\\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \\leq n < 2(s+2+\\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\\in(0,1)$ in dimensions $2 \\leq n\\leq 6$. It applies to all nonlinearities $f\\in C^2$. For such parameters $s$ and $n$, our result leads to the regularity of the extremal solution when $f$ is replaced by $\\lambda f$ with $\\lambda > 0$. This is a widely studied question for $s=1$, which is still largely open in the nonradial case both for $s=1$ and $s<1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-22T17:55:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J61",
      "35R11",
      "35B45",
      "35B35",
      "35B65",
      "35J70"
    ],
    "keywords": [
      "semilinear elliptic equations",
      "radial stable solutions",
      "fractional laplacian",
      "regularity",
      "main result establishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}