{
  "id": "2105.10363",
  "version": "v1",
  "published": "2021-05-21T14:00:36.000Z",
  "updated": "2021-05-21T14:00:36.000Z",
  "title": "Classification of Positive Radial Solutions to A Weighted Biharmonic Equation",
  "authors": [
    "Yuhao Yan"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the weighted fourth order equation $$\\Delta(|x|^{-\\alpha}\\Delta u)+\\lambda \\text{div}(|x|^{-\\alpha-2}\\nabla u)+\\mu|x|^{-\\alpha-4}u=|x|^\\beta u^p\\quad \\text{in} \\quad \\mathbb{R}^n \\backslash \\{0\\},$$ where $n\\geq 5$, $-n<\\alpha<n-4$, $p>1$ and $(p,\\alpha,\\beta,n)$ belongs to the critical hyperbola $$\\frac{n+\\alpha}{2}+\\frac{n+\\beta}{p+1}=n-2.$$ We prove the existence of radial solutions to the equation for some $\\lambda$ and $\\mu$. On the other hand, let $v(t):=|x|^{\\frac{n-4-\\alpha}{2}}u(|x|)$, $t=-\\ln |x|$, then for the radial solution $u$ with non-removable singularity at origin, $v(t)$ is a periodic function if $\\alpha \\in (-2,n-4)$ and $\\lambda$, $\\mu$ satisfy some conditions; while for $\\alpha \\in (-n,-2]$, there exists a radial solution with non-removable singularity and the corresponding function $v(t)$ is not periodic. We also get some results about the best constant and symmetry breaking, which is closely related to the Caffarelli-Kohn-Nirenberg type inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-21T14:00:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B07",
      "35B09",
      "35B10",
      "35J30",
      "35J75"
    ],
    "keywords": [
      "weighted biharmonic equation",
      "positive radial solutions",
      "classification",
      "weighted fourth order equation",
      "non-removable singularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}