{
  "id": "2208.13272",
  "version": "v1",
  "published": "2022-08-28T19:07:05.000Z",
  "updated": "2022-08-28T19:07:05.000Z",
  "title": "Uniqueness of entire solutions to quasilinear equations of p-Laplace type",
  "authors": [
    "Nguyen Cong Phuc",
    "Igor E. Verbitsky"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove the uniqueness property for a class of entire solutions to the equation \\begin{equation*} \\left\\{ \\begin{array}{ll} -{\\rm div}\\, \\mathcal{A}(x,\\nabla u) = \\sigma, \\quad u\\geq 0 \\quad \\text{in } \\mathbb{R}^n, \\\\ \\displaystyle{\\liminf_{|x|\\rightarrow \\infty}}\\, u = 0, \\end{array} \\right. \\end{equation*} where $\\sigma$ is a nonnegative locally finite measure in $\\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and ${\\rm div}\\, \\mathcal{A}(x,\\nabla u)$ is the $\\mathcal{A}$-Laplace operator, under standard growth and monotonicity assumptions of order $p$ ($1<p<\\infty$) on $\\mathcal{A}(x, \\xi)$ ($x, \\xi \\in \\mathbb{R}^n$); the model case $\\mathcal{A}(x, \\xi)=\\xi | \\xi |^{p-2}$ corresponds to the $p$-Laplace operator $\\Delta_p$ on $\\mathbb{R}^n$. We also establish uniqueness of solutions to a similar problem, \\begin{equation*} \\left\\{ \\begin{array}{ll} -{\\rm div}\\, \\mathcal{A}(x,\\nabla u) = \\sigma u^q +\\mu, \\quad u\\geq 0 \\quad \\text{in } \\mathbb{R}^n, \\\\ \\displaystyle{\\liminf_{|x|\\rightarrow \\infty}}\\, u = 0, \\end{array} \\right. \\end{equation*} in the sub-natural growth case $0<q<p-1$, where $\\mu, \\sigma$ are nonnegative locally finite measures in $\\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and $\\mathcal{A}(x, \\xi)$ satisfies an additional homogeneity condition, which holds in particular for the $p$-Laplace operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-28T19:07:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "31B15",
      "42B37"
    ],
    "keywords": [
      "entire solutions",
      "quasilinear equations",
      "p-laplace type",
      "laplace operator",
      "nonnegative locally finite measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}