{
  "id": "1011.4069",
  "version": "v1",
  "published": "2010-11-17T21:01:35.000Z",
  "updated": "2010-11-17T21:01:35.000Z",
  "title": "Positive Solutions for the p-Laplacian with Dependence on the Gradient",
  "authors": [
    "Hamilton Bueno",
    "Grey Ercole",
    "Wenderson Ferreira",
    "Antônio Zumpano"
  ],
  "journal": "Nonlinearity 25 (2012) 1211-1234",
  "doi": "10.1088/0951-7715/25/4/1211",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove a result of existence of positive solutions of the Dirichlet problem for $-\\Delta_p u=\\mathrm{w}(x)f(u,\\nabla u)$ in a bounded domain $\\Omega\\subset\\mathbb{R}^N$, where $\\Delta_p$ is the $p$-Laplacian and $\\mathrm{w}$ is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on $f$, but simple geometric assumptions on a neighborhood of the first eigenvalue of the $p$-Laplacian operator. We start by solving the problem in a radial domain by applying the Schauder Fixed Point Theorem and this result is used to construct an ordered pair of sub- and super-solution, also valid for nonlinearities which are super-linear both at the origin and at $+\\infty$. We apply our method to the Dirichlet problem $-\\Delta_pu = \\lambda u(x)^{q-1}(1+|\\nabla u(x)|^p)$ in $\\Omega$ and give examples of super-linear nonlinearities which are also handled by our method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-17T21:01:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive solutions",
      "p-laplacian",
      "dependence",
      "dirichlet problem",
      "schauder fixed point theorem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Nonlinearity",
      "year": 2012,
      "month": "Apr",
      "volume": 25,
      "number": 4,
      "pages": 1211
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012Nonli..25.1211B"
    }
  }
}