{
  "id": "math/0512149",
  "version": "v1",
  "published": "2005-12-07T11:05:35.000Z",
  "updated": "2005-12-07T11:05:35.000Z",
  "title": "Concentration phenomena for a fourth order equations with exponential growth: the radial case",
  "authors": [
    "Frederic Robert"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We let $\\Omega$ be a smooth bounded domain of $\\mathbb{R}^4$ and a sequence of fonctions $(V_k)_{k\\in\\mathbb{N}}\\in C^0(\\Omega)$ such that $\\lim_{k\\to +\\infty}V_k=1$ in $C^0_{loc}(\\Omega)$. We consider a sequence of functions $(u_k)_{k\\in\\mathbb{N}}\\in C^4(\\Omega)$ such that $$\\Delta^2 u_k=V_k e^{4u_k}$$ in $\\Omega$ for all $k\\in\\mathbb{N}$. We address in this paper the question of the asymptotic behaviour of the $(u_k)'s$ when $k\\to +\\infty$. The corresponding problem in dimension 2 was considered by Br\\'ezis-Merle and Li-Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author, a similar quantization phenomenon does not hold for this fourth order problem. Assuming that the $u_k$'s are radially symmetrical, we push further the previous analysis. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-12-07T11:05:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35J35"
    ],
    "keywords": [
      "fourth order equations",
      "exponential growth",
      "concentration phenomena",
      "radial case",
      "similar quantization phenomenon"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/j.jde.2006.03.019",
      "journal": "Journal of Differential Equations",
      "year": 2006,
      "volume": 231,
      "number": 1,
      "pages": 135
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006JDE...231..135R"
    }
  }
}