{
  "id": "2008.07211",
  "version": "v1",
  "published": "2020-08-17T10:48:43.000Z",
  "updated": "2020-08-17T10:48:43.000Z",
  "title": "Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms",
  "authors": [
    "Caihong Chang",
    "Zhengce Zhang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show the Liouville-type theorems for positive weak solutions of the equation involving the $m$-Laplacian operator \\begin{equation*} -\\Delta_{m}u=u^q|\\nabla u|^p\\ \\ \\ \\mathrm{in}\\ \\mathbb{R}^N, \\end{equation*} where $N\\geq1$, $m>1$ and $p,q\\geq0$. This paper mainly adopts the technique of Bernstein gradient estimates to study from three cases: $p>m$, $p=m$ and $p<m$, respectively. Meanwhile, this paper also obtains a Liouville-type theorem for supersolutions under certain assumptions on $m, p, q$ and $N$. Then, as its applications, we apply a degree argument to obtain the existence of positive weak solutions for a nonlinear Dirichlet problem. Our proof is based on a priori estimates, which will be accomplished by using a blow-up argument together with the Liouville-type theorem. Meanwhile, some new Harnack inequalities are proved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-17T10:48:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasilinear elliptic equations",
      "nonlinear gradient terms",
      "liouville-type theorem",
      "positive weak solutions",
      "bernstein gradient estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}