{
  "id": "1709.05544",
  "version": "v1",
  "published": "2017-09-16T18:11:12.000Z",
  "updated": "2017-09-16T18:11:12.000Z",
  "title": "Existence result under flatness condition for a nonlinear elliptic equation with Sobolev exponent",
  "authors": [
    "Zakaria Boucheche"
  ],
  "comment": "29 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the following nonlinear elliptic equation with Dirichlet boundary condition: $-\\Delta u=K(x)u^{\\frac{n+2}{n-2}},\\, u>0$ in $\\Omega,\\, u=0$ on $\\partial\\Omega$, where $\\Omega$ is a smooth bounded domain in $\\mathbb{R}^n,$ $n\\geqslant 4,$ and $K$ is a $\\mathcal{C}^1$-positive function in $\\bar{\\Omega}$. Under the assumption that the order of flatness at each critical point of $K$ is $\\beta \\in ]\\,n-2,\\,n[,$ we give precise estimates on the looses of the compactness, and we prove an existence result through an Euler-Hopf type formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-16T18:11:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60"
    ],
    "keywords": [
      "nonlinear elliptic equation",
      "existence result",
      "flatness condition",
      "sobolev exponent",
      "dirichlet boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}