{
  "id": "1211.2223",
  "version": "v2",
  "published": "2012-11-09T20:29:30.000Z",
  "updated": "2012-11-20T11:01:17.000Z",
  "title": "On stable solutions of biharmonic problem with polynomial growth",
  "authors": [
    "Hatem Hajlaoui",
    "Abdelaziz Harrabi",
    "Dong Ye"
  ],
  "comment": "Corrected typos",
  "journal": "Pacific J. Maths, vol. 270(1), pp 79-93, 2014",
  "doi": "10.2140/pjm.2014.270.79",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove the nonexistence of smooth stable solution to the biharmonic problem $\\Delta^2 u= u^p$, $u>0$ in $\\R^N$ for $1 < p < \\infty$ and $N < 2(1 + x_0)$, where $x_0$ is the largest root of the following equation: $$x^4 - \\frac{32p(p+1)}{(p-1)^2}x^2 + \\frac{32p(p+1)(p+3)}{(p-1)^3}x -\\frac{64p(p+1)^2}{(p-1)^4} = 0.$$ In particular, as $x_0 > 5$ when $p > 1$, we obtain the nonexistence of smooth stable solution for any $N \\leq 12$ and $p > 1$. Moreover, we consider also the corresponding problem in the half space $\\R^N_+$, or the elliptic problem $\\Delta^2 u= \\l(u+1)^p$ on a bounded smooth domain $\\O$ with the Navier boundary conditions. We will prove the regularity of the extremal solution in lower dimensions. Our results improve the previous works.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-11-20T11:01:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J91",
      "35J30",
      "35J40"
    ],
    "keywords": [
      "biharmonic problem",
      "polynomial growth",
      "smooth stable solution",
      "navier boundary conditions",
      "bounded smooth domain"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.2223H"
    }
  }
}