{
  "id": "2007.00069",
  "version": "v1",
  "published": "2020-06-30T19:22:59.000Z",
  "updated": "2020-06-30T19:22:59.000Z",
  "title": "On stable and finite Morse index solutions of the fractional Toda system",
  "authors": [
    "Mostafa Fazly",
    "Wen Yang"
  ],
  "comment": "21 pages. Comments are welcome",
  "categories": [
    "math.AP"
  ],
  "abstract": "We develop a monotonicity formula for solutions of the fractional Toda system $$ (-\\Delta)^s f_\\alpha = e^{-(f_{\\alpha+1}-f_\\alpha)} - e^{-(f_\\alpha-f_{\\alpha-1})} \\quad \\text{in} \\ \\ \\mathbb R^n,$$ when $0<s<1$, $\\alpha=1,\\cdots,Q$, $f_0=-\\infty$, $f_{Q+1}=\\infty$ and $Q \\ge2$ is the number of equations in this system. We then apply this formula, technical integral estimates, classification of stable homogeneous solutions, and blow-down analysis arguments to establish Liouville type theorems for finite Morse index (and stable) solutions of the above system when $n > 2s$ and $$ \\dfrac{\\Gamma(\\frac{n}{2})\\Gamma(1+s)}{\\Gamma(\\frac{n-2s}{2})} \\frac{Q(Q-1)}{2} > \\frac{ \\Gamma(\\frac{n+2s}{4})^2 }{ \\Gamma(\\frac{n-2s}{4})^2} . $$ Here, $\\Gamma$ is the Gamma function. When $Q=2$, the above equation is the classical (fractional) Gelfand-Liouville equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-30T19:22:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite morse index solutions",
      "fractional toda system",
      "establish liouville type theorems",
      "blow-down analysis arguments",
      "technical integral estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}