{
  "id": "1802.00109",
  "version": "v1",
  "published": "2018-02-01T00:32:28.000Z",
  "updated": "2018-02-01T00:32:28.000Z",
  "title": "Gradient estimates for nonlinear elliptic equations with a gradient-dependent nonlinearity",
  "authors": [
    "Joshua Ching",
    "Florica C. Cirstea"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we obtain gradient estimates of the positive solutions to weighted $p$-Laplacian type equations with a gradient-dependent nonlinearity of the form \\begin{equation} \\label{one} {\\rm div} (|x|^{\\sigma}|\\nabla u|^{p-2} \\nabla u)= |x|^{-\\tau} u^q |\\nabla u|^m \\quad \\mbox{in } \\ \\Omega^*:= \\Omega \\setminus \\{ 0 \\}. \\end{equation} Here, $\\Omega\\subseteq \\mathbb R^N$ denotes a domain containing the origin with $N\\geq 2$, whereas $m,q\\in [0,\\infty)$, $1<p\\leq N+\\sigma$ and $q>\\max\\{p-m-1,\\sigma+\\tau-1\\}$. The main difficulty arises from the dependence of the right-hand side of the equation on $x$, $u$ and $|\\nabla u|$, without any upper bound restriction on the power $m$ of $|\\nabla u|$. Our proof of the gradient estimates is based on a two-step process relying on a modified version of the Bernstein's method. As a by-product, we extend the range of applicability of the Liouville-type results known for our problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-01T00:32:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35B53"
    ],
    "keywords": [
      "nonlinear elliptic equations",
      "gradient estimates",
      "gradient-dependent nonlinearity",
      "laplacian type equations",
      "main difficulty arises"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}