{
  "id": "2310.08119",
  "version": "v1",
  "published": "2023-10-12T08:22:57.000Z",
  "updated": "2023-10-12T08:22:57.000Z",
  "title": "The existence of ground state solutions for nonlinear p-Laplacian equations on lattice graphs",
  "authors": [
    "Bobo Hua",
    "Wendi Xu"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP",
    "math.CO"
  ],
  "abstract": "In this paper, we study the nonlinear $p$-Laplacian equation $$-\\Delta_{p} u+V(x)|u|^{p-2}u=f(x,u) $$ with positive and periodic potential $V$ on the lattice graph $\\mathbb{Z}^{N}$, where $\\Delta_{p}$ is the discrete $p$-Laplacian, $p \\in (1,\\infty)$. The nonlinearity $f$ is also periodic in $x$ and satisfies the growth condition $|f(x,u)| \\leq a(1+|u|^{q-1})$ for some $ q>p$. We first prove the equivalence of three function spaces on $\\mathbb{Z}^{N}$, which is quite different from the continuous case and allows us to remove the restriction $q>p^{*}$ in [SW10], where $p^{*}$ is the critical exponent for $ W^{1,p}(\\Omega) \\hookrightarrow L^{q}(\\Omega)$ with $\\Omega \\subset \\mathbb{R}^{N}$ bounded. Then, using the method of Nehari [Neh60, Neh61], we prove the existence of ground state solutions to the above equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-12T08:22:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "39A14",
      "58E30"
    ],
    "keywords": [
      "ground state solutions",
      "nonlinear p-laplacian equations",
      "lattice graph",
      "periodic potential",
      "growth condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}