{
  "id": "1709.03070",
  "version": "v1",
  "published": "2017-09-10T08:47:03.000Z",
  "updated": "2017-09-10T08:47:03.000Z",
  "title": "Existence of positive solutions to a nonlinear elliptic system with nonlinearity involving gradient term",
  "authors": [
    "Boumediene Abdellaoui",
    "Ahmed Attar",
    "El-Haj Laamri"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work we analyze the existence of solutions to the nonlinear elliptic system: \\begin{equation*} \\left\\{ \\begin{array}{rcll} -\\Delta u & = & v^q+\\a g & \\text{in }\\Omega , \\\\ -\\Delta v& = &|\\nabla u|^{p}+\\l f &\\text{in }\\Omega , \\\\ u=v&=& 0 & \\text{on }\\partial \\Omega ,\\\\ u,v& \\geq & 0 & \\text{in }\\Omega, \\end{array}% \\right. \\end{equation*} where $\\Omega$ is a bounded domain of $\\ren$ and $p\\ge 1$, $q>0$ with $pq>1$. $f,g$ are nonnegative measurable functions with additional hypotheses and $\\a, \\l\\ge 0$. As a consequence we show that the fourth order problem \\begin{equation*} \\left\\{ \\begin{array}{rcll} \\Delta^2 u & = &|\\nabla u|^{p}+\\tilde{\\l} \\tilde{f} &\\text{in }\\Omega , \\\\ u=\\D u&=& 0 & \\text{on }\\partial \\Omega ,\\\\ \\end{array}% \\right. \\end{equation*} has a solution for all $p>1$, under suitable conditions on $\\tilde{f}$ and $\\tilde{\\l}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-10T08:47:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J55",
      "35D10",
      "35J60"
    ],
    "keywords": [
      "nonlinear elliptic system",
      "gradient term",
      "positive solutions",
      "nonlinearity",
      "fourth order problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}