{
  "id": "1405.0162",
  "version": "v2",
  "published": "2014-05-01T14:06:37.000Z",
  "updated": "2014-10-15T05:49:14.000Z",
  "title": "A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities",
  "authors": [
    "Mousomi Bhakta"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence and non-existence of nontrivial weak solution of $$ {\\Delta^2u-\\mu\\frac{u}{|x|^{4}} = \\frac{|u|^{q_{\\beta}-2}u}{|x|^{\\beta}}+|u|^{q-2}u\\quad\\textrm{in ${\\mathbb R}^N$,}} $$ where $N\\geq 5$, $q_{\\beta}=\\frac{2(N-\\beta)}{N-4}$, $0<\\beta<4$, $1<q\\leq 2^{**}$ and $\\mu<\\mu_1:=\\big(\\frac{N(N-4)}{4}\\big)^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q< 2^{**}$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{**}$ and $-(N-2)^2\\leq\\mu<\\mu_1$, we establish the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti and Rabinowitz and the variational methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-01T14:06:37.000Z",
      "comment": "13 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-15T05:49:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B30",
      "35J35",
      "35J91"
    ],
    "keywords": [
      "multiple critical nonlinearities",
      "semilinear elliptic equation",
      "biharmonic operator",
      "nontrivial weak solution",
      "non-existence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0162B"
    }
  }
}