{
  "id": "1803.11520",
  "version": "v2",
  "published": "2018-03-30T15:50:53.000Z",
  "updated": "2024-10-26T03:29:22.000Z",
  "title": "A complete description of the asymptotic behavior at infinity of positive radial solutions to $Δ^2 u = u^α$ in $\\mathbf R^n$",
  "authors": [
    "Quôc Anh Ngô",
    "Van Hoang Nguyen",
    "Quoc Hung Phan"
  ],
  "comment": "27 pages, 0 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the biharmonic equation $\\Delta^2 u = u^\\alpha$ in $\\mathbf R^n$ with $n \\geqslant 1$. It was proved that this equation has a positive classical solution if, and only if, either $\\alpha \\leqslant 1$ with $n \\geqslant 1$ or $\\alpha\\geqslant (n+4)/(n-4)$ with $n \\geqslant 5$. The asymptotic behavior at infinity of all positive radial solutions was known in the case $\\alpha\\geqslant (n+4)/(n-4)$ and $n \\geqslant 5$. In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case $\\alpha\\leqslant 1$ with $n \\geqslant 1$; hence obtaining a complete picture of the asymptotic behavior at infinity of positive radial solutions. Since the underlying equation is higher order, we propose a new approach which relies on a representation formula and asymptotic analysis arguments. We believe that the approach introduced here can be conveniently applied to study other problems with higher order operators.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-10-26T03:29:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B08",
      "35B40",
      "35J91",
      "35B09",
      "35B51"
    ],
    "keywords": [
      "positive radial solutions",
      "asymptotic behavior",
      "complete description",
      "higher order operators",
      "asymptotic analysis arguments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}