{
  "id": "1508.06006",
  "version": "v1",
  "published": "2015-08-25T01:49:35.000Z",
  "updated": "2015-08-25T01:49:35.000Z",
  "title": "On critical $p$-Laplacian systems",
  "authors": [
    "Zhenyu Guo",
    "Kanishka Perera",
    "Wenming Zou"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the critical $p$-Laplacian system \\begin{equation}\\label{92} \\begin{cases}-\\Delta_p u-\\frac{\\lambda a}{p}|u|^{a-2}u|v|^b =\\mu_1|u|^{p^\\ast-2}u+\\frac{\\alpha\\gamma}{p^\\ast}|u|^{\\alpha-2}u|v|^{\\beta}, &x\\in\\Omega,\\\\ -\\Delta_p v-\\frac{\\lambda b}{p}|u|^a|v|^{b-2}v =\\mu_2|v|^{p^\\ast-2}v+\\frac{\\beta\\gamma}{p^\\ast}|u|^{\\alpha}|v|^{\\beta-2}v, &x\\in\\Omega,\\\\ u,v\\ \\text{in } D_0^{1,p}(\\Omega), \\end{cases} \\end{equation} where $\\Delta_p:=\\text{div}(|\\nabla u|^{p-2}\\nabla u)$ is the $p$-Laplacian operator defined on $D^{1,p}(\\mathbb{R}^N):=\\{u\\in L^{p^\\ast}(\\mathbb{R}^N):|\\nabla u|\\in L^p(\\mathbb{R}^N)\\}$, endowed with norm $\\|u\\|_{D^{1,p}}:=\\big(\\int_{\\mathbb{R}^N}|\\nabla u|^p\\text{d}x\\big)^{\\frac{1}{p}}$, $N\\ge3$, $1<p<N$, $\\lambda, \\mu_1, \\mu_2\\ge 0$, $\\gamma\\neq0$, $a, b, \\alpha, \\beta > 1$ satisfy $a + b = p, \\alpha + \\beta = p^\\ast:=\\frac{Np}{N-p}$, the critical Sobolev exponent, $\\Omega$ is $\\mathbb{R}^N$ or a bounded domain in $\\mathbb{R}^N$, $D_0^{1,p}(\\Omega)$ is the closure of $C_0^\\infty(\\Omega)$ in $D^{1,p}(\\mathbb{R}^N)$. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution. We also consider the existence and multiplicity of nontrivial nonnegative solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-25T01:49:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplacian system",
      "critical sobolev exponent",
      "nontrivial nonnegative solutions",
      "laplacian operator",
      "energy solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150806006G"
    }
  }
}