{
  "id": "math/0512148",
  "version": "v1",
  "published": "2005-12-07T10:54:21.000Z",
  "updated": "2005-12-07T10:54:21.000Z",
  "title": "Quantization effects for a fourth order equation of exponential growth in dimension four",
  "authors": [
    "Frederic Robert"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate the asymptotic behavior as $k \\to +\\infty$ of sequences $(u_k)_{k\\in\\mathbb{N}}\\in C^4(\\Omega)$ of solutions of the equations $\\Delta^2 u_k=V_k e^{4u_k}$ on $\\Omega$, where $\\Omega$ is a bounded domain of $\\mathbb{R}^4$ and $\\lim_{k\\to +\\infty}V_k=1$ in $C^0_{loc}(\\Omega)$. The corresponding 2-dimensional problem was studied by Br\\'ezis-Merle and Li-Shafrir who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Struwe and the author, such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on $\\Delta u_k$, we recover such a quantization as in dimension 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-12-07T10:54:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35J35"
    ],
    "keywords": [
      "fourth order equation",
      "exponential growth",
      "quantization effects",
      "natural hypothesis",
      "blow-up occurs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12148R"
    }
  }
}