{
  "id": "1206.6233",
  "version": "v1",
  "published": "2012-06-27T11:30:50.000Z",
  "updated": "2012-06-27T11:30:50.000Z",
  "title": "Boundedness of the extremal solutions in dimension 4",
  "authors": [
    "Salvador Villegas"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we establish the boundedness of the extremal solution u^* in dimension N=4 of the semilinear elliptic equation $-\\Delta u=\\lambda f(u)$, in a general smooth bounded domain Omega of R^N, with Dirichlet data $u|_{\\partial \\Omega}=0$, where f is a C^1 positive, nondecreasing and convex function in [0,\\infty) such that $f(s)/s\\rightarrow\\infty$ as $s\\rightarrow\\infty$. In addition, we prove that, for N>=5, the extremal solution $u^*\\in W^{2,\\frac{N}{N-2}}$. This gives $u^\\ast\\in L^\\frac{N}{N-4}$, if N>=5 and $u^*\\in H_0^1$, if N=6.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-27T11:30:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extremal solution",
      "boundedness",
      "general smooth bounded domain omega",
      "semilinear elliptic equation",
      "dirichlet data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.6233V"
    }
  }
}