{
  "id": "2008.07374",
  "version": "v1",
  "published": "2020-08-17T14:39:58.000Z",
  "updated": "2020-08-17T14:39:58.000Z",
  "title": "On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand-Liouville equation",
  "authors": [
    "Mostafa Fazly",
    "Yeyao Hu",
    "Wen Yang"
  ],
  "comment": "24 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2003.03071",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the nonlocal H\\'{e}non-Gelfand-Liouville problem $$ (-\\Delta)^s u = |x|^a e^u\\quad\\mathrm{in}\\quad \\mathbb R^n, $$ for every $s\\in(0,1)$, $a>0$ and $n>2s$. We prove a monotonicity formula for solutions of the above equation using rescaling arguments. We apply this formula together with blow-down analysis arguments and technical integral estimates to establish non-existence of finite Morse index solutions when $$\\dfrac{\\Gamma(\\frac n2)\\Gamma(s)}{\\Gamma(\\frac{n-2s}{2})}\\left(s+\\frac a2\\right)> \\dfrac{\\Gamma^2(\\frac{n+2s}{4})}{\\Gamma^2(\\frac{n-2s}{4})}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-17T14:39:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite morse index solutions",
      "nonlocal hénon-gelfand-liouville equation",
      "blow-down analysis arguments",
      "monotonicity formula",
      "technical integral estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}