{
  "id": "2208.10794",
  "version": "v1",
  "published": "2022-08-23T08:00:36.000Z",
  "updated": "2022-08-23T08:00:36.000Z",
  "title": "Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient type",
  "authors": [
    "Anna Maria Candela",
    "Addolorata Salvatore",
    "Caterina Sportelli"
  ],
  "journal": "Adv. Nonlinear Stud. 21 (2021) 461-488",
  "doi": "10.1515/ans-2021-2121",
  "categories": [
    "math.AP"
  ],
  "abstract": "The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \\[ (P)\\qquad \\left\\{ \\begin{array}{ll} - {\\rm div} (A(x, u)\\vert\\nabla u\\vert^{p_1 -2} \\nabla u) + \\frac{1}{p_1}A_u (x, u)\\vert\\nabla u\\vert^{p_1} = G_u(x, u, v) &\\hbox{ in $\\Omega$,}\\\\[5pt] - {\\rm div} (B(x, v)\\vert\\nabla v\\vert^{p_2 -2} \\nabla v) +\\frac{1}{p_2}B_v(x, v)\\vert\\nabla v\\vert^{p_2} = G_v\\left(x, u, v\\right) &\\hbox{ in $\\Omega$,}\\\\[5pt] u = v = 0 &\\hbox{ on $\\partial\\Omega$,} \\end{array} \\right. \\] where $\\Omega \\subset \\mathbb{R}^N$ is an open bounded domain, $p_1$, $p_2 > 1$ and $A(x,u)$, $B(x,v)$ are $\\mathcal{C}^1$-Carath\\'eodory functions on $\\Omega \\times \\mathbb{R}$ with partial derivatives $A_u(x,u)$, respectively $B_v(x,v)$, while $G_u(x,u,v)$, $G_v(x,u,v)$ are given Carath\\'eodory maps defined on $\\Omega \\times \\mathbb{R}\\times \\mathbb{R}$ which are partial derivatives of a function $G(x,u,v)$. We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional $\\cal{J}$, related to problem $(P)$, admits at least one critical point in the ''right'' Banach space $X$. Moreover, if $\\cal{J}$ is even, then $(P)$ has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami-Palais-Smale condition, a ''good'' decomposition of the Banach space $X$ and suitable generalizations of the Ambrosetti-Rabinowitz Mountain Pass Theorems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-23T08:00:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J50",
      "35J92",
      "47J30",
      "58E05"
    ],
    "keywords": [
      "coupled quasilinear elliptic system",
      "gradient type",
      "multiplicity results",
      "ambrosetti-rabinowitz mountain pass theorems",
      "partial derivatives"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}