{
  "id": "1710.03440",
  "version": "v1",
  "published": "2017-10-10T08:13:53.000Z",
  "updated": "2017-10-10T08:13:53.000Z",
  "title": "Multiplicity of solutions for a class of elliptic problem of $p$-Laplacian type with a $p$-Gradient term",
  "authors": [
    "Zakariya Chaouai",
    "Soufiane Maatouk"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the following problem $$(P) \\begin{cases} -\\Delta_{p}u= c(x)|u|^{q-1}u+\\mu |\\nabla u|^{p}+h(x) & \\ \\ \\mbox{ in }\\Omega, u=0 & \\ \\ \\mbox{ on } \\partial\\Omega, \\end{cases}$$ where $\\Omega$ is a bounded set in $\\mathbb{R}^{N}$ ($N\\geq 3$) with a smooth boundary, $1<p<N$, $q>0$, $\\mu \\in \\mathbb{R}^{*}$ and $c, h$ belongs to $L^{k}(\\Omega)$ for some $k>\\frac{N}{p}$. In this paper, we assume that $c$ and $h$ are allowed to change sign such that $c^{+}\\not \\equiv 0$, then we prove the existence of at least two bounded solutions when $\\|c\\|_{k}$ and $\\|h\\|_{k}$ are suitably small. For this purpose, we use the Mountain Pass theorem, due to Ambrosetti and Rabinowitz, on a problem with variational structure reduced from $(P)$. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the Ambrosetti and Rabinowitz condition by the \\textbf{nonquadraticity condition at infinity}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-10T08:13:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic problem",
      "laplacian type",
      "gradient term",
      "multiplicity",
      "rabinowitz condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}