{
  "id": "1406.7681",
  "version": "v2",
  "published": "2014-06-30T11:40:51.000Z",
  "updated": "2015-02-10T10:15:10.000Z",
  "title": "A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions",
  "authors": [
    "Alessandro Iacopetti",
    "Filomena Pacella"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Brezis-Nirenberg problem: \\begin{equation*} \\begin{cases} -\\Delta u = \\lambda u + |u|^{2^* -2}u & \\hbox{in}\\ \\Omega\\\\ u=0 & \\hbox{on}\\ \\partial \\Omega, \\end{cases} \\end{equation*} where $\\Omega$ is a smooth bounded domain in $\\mathbb{R}^N$, $N\\geq 3$, $2^{*}=\\frac{2N}{N-2}$ is the critical Sobolev exponent and $\\lambda>0$ a positive parameter. The main result of the paper shows that if $N=4,5,6$ and $\\lambda$ is close to zero there are no sign-changing solutions of the form $$u_\\lambda=PU_{\\delta_1,\\xi}-PU_{\\delta_2,\\xi}+w_\\lambda, $$ where $PU_{\\delta_i}$ is the projection on $H_0^1(\\Omega)$ of the regular positive solution of the critical problem in $\\mathbb{R}^N$, centered at a point $\\xi \\in \\Omega$ and $w_\\lambda$ is a remainder term. Some additional results on norm estimates of $w_\\lambda$ and about the concentrations speeds of tower of bubbles in higher dimensions are also presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-30T11:40:51.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-10T10:15:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brezis-nirenberg problem",
      "sign-changing solutions",
      "nonexistence result",
      "low dimensions",
      "smooth bounded domain"
    ],
    "publication": {
      "doi": "10.1016/j.jde.2015.01.030",
      "journal": "Journal of Differential Equations",
      "year": 2015,
      "month": "Jun",
      "volume": 258,
      "number": 12,
      "pages": 4180
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JDE...258.4180I"
    }
  }
}