{
  "id": "2410.11659",
  "version": "v1",
  "published": "2024-10-15T14:45:58.000Z",
  "updated": "2024-10-15T14:45:58.000Z",
  "title": "A quasilinear elliptic equation with absorption term and Hardy potential",
  "authors": [
    "Marie-Françoise Bidaut-Véron Huyuan Chen"
  ],
  "comment": "47 pages,4 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "Here we study the positive solutions of the equation \\begin{equation*} -\\Delta _{p}u+\\mu \\frac{u^{p-1}}{\\left\\vert x\\right\\vert ^{p}}+\\left\\vert x\\right\\vert ^{\\theta }u^{q}=0,\\qquad x\\in \\mathbb{R}^{N}\\backslash \\left\\{ 0\\right\\} \\end{equation*}% where $\\Delta _{p}u={div}(\\left\\vert \\nabla u\\right\\vert ^{p-2}\\nabla u) $ and $1<p<N,q>p-1,\\mu ,\\theta \\in \\mathbb{R}.$ We give a complete description of the existence and the asymptotic behaviour of the solutions near the singularity $0,$ or in an exterior domain. We show that the global solutions $\\mathbb{R}^{N}\\backslash \\left\\{ 0\\right\\} $ are radial and give their expression according to the position of the Hardy coefficient $\\mu $ with respect to the critical exponent $\\mu _{0}=-(\\frac{N-p}{p})^{p}.$ Our method consists into proving that any nonradial solution can be compared to a radial one, then making exhaustive radial study by phase-plane techniques. Our results are optimal, extending the known results when $\\mu =0$ or $p=2$, with new simpler proofs.They make in evidence interesting phenomena of nonuniqueness when $\\theta +p=0$, and of existence of locally constant solutions when moreover $p>2$ .",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-15T14:45:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "35J75"
    ],
    "keywords": [
      "quasilinear elliptic equation",
      "absorption term",
      "hardy potential",
      "evidence interesting phenomena",
      "complete description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}