{
  "id": "1909.11918",
  "version": "v1",
  "published": "2019-09-26T06:09:53.000Z",
  "updated": "2019-09-26T06:09:53.000Z",
  "title": "Gradient-type systems on unbounded domains of the Heisenberg group",
  "authors": [
    "Giovanni Molica Bisci",
    "Dušan D. Repovš"
  ],
  "journal": "J. Geom. Anal. (2019). https://doi.org/10.1007/s12220-019-00276-2",
  "doi": "10.1007/s12220-019-00276-2",
  "categories": [
    "math.AP"
  ],
  "abstract": "The purpose of this paper is to study the existence of weak solutions for some classes of one-parameter subelliptic gradient-type systems involving a Sobolev-Hardy potential defined on an unbounded domain $\\Omega_\\psi$ of the Heisenberg group $\\mathbb{H}^n=\\mathbb{C}^n\\times \\mathbb{R}$ ($n\\geq 1$) whose geometrical profile is determined by two real positive functions $\\psi_1$ and $\\psi_2$ that are bounded on bounded sets. The treated problems have a variational structure and thanks to this, we are able to prove the existence of an open interval $\\Lambda\\subset (0,\\infty)$ such that, for every parameter $\\lambda\\in \\Lambda$, the system has at least two nontrivial symmetric weak solutions that are uniformly bounded with respect to the Sobolev $HW^{1,2}_0$-norm. Moreover, the existence is stable under certain small subcritical perturbations of the nonlinear term. The main proof, crucially based on the Palais principle of symmetric criticality, is obtained by developing a group-theoretical procedure on the unitary group $\\mathbb{U}(n)=U(n)\\times\\{1\\}$ and by exploiting some compactness embedding results into Lebesgue spaces, recently proved for suitable $\\mathbb{U}(n)$-invariant subspaces of the Folland-Stein space $HW^{1,2}_0(\\Omega_\\psi)$. A key ingredient for our variational approach is a very general min-max argument valid for sufficiently smooth functionals defined on reflexive Banach spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-26T06:09:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R03",
      "35H20",
      "35J70",
      "35A01",
      "35A15"
    ],
    "keywords": [
      "heisenberg group",
      "unbounded domain",
      "one-parameter subelliptic gradient-type systems",
      "general min-max argument valid",
      "nontrivial symmetric weak solutions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}