{
  "id": "1602.03354",
  "version": "v1",
  "published": "2016-02-10T12:45:34.000Z",
  "updated": "2016-02-10T12:45:34.000Z",
  "title": "On the Topological degree of the Mean field equation with two parameters",
  "authors": [
    "Aleks Jevnikar",
    "Juncheng Wei",
    "Wen Yang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $$ - \\Delta u = \\rho_1 \\left( \\frac{h_1 \\,e^{u}}{\\int_M h_1 \\,e^{u} } - \\frac{1}{|M|} \\right) - \\rho_2 \\left( \\frac{h_2 \\,e^{-u}}{\\int_M h_2 \\,e^{-u} } - \\frac{1}{|M|} \\right), $$ which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h_1, h_2$ are smooth positive functions and $\\rho_1, \\rho_2$ are two positive parameters. We start by proving a concentration phenomena for the above equation, which leads to a-priori bound for the solutions of this problem provided $\\rho_i\\notin 8\\pi\\mathbb{N}, \\, i=1,2$. Then we study the blow up behavior when $\\rho_1$ crosses $8\\pi$ and $\\rho_2 \\notin 8\\pi\\mathbb{N}$. By performing a suitable decomposition of the above equation and using the shadow system that was introduced for the $SU(3)$ Toda system, we can compute the Leray-Schauder topological degree for $\\rho_1 \\in (0,8\\pi) \\cup (8\\pi,16\\pi)$ and $\\rho_2 \\notin 8\\pi\\mathbb{N}$. As a byproduct our argument, we give new existence results when the underlying manifold is a sphere and a new proof for some known existence result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-10T12:45:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J60",
      "35R01"
    ],
    "keywords": [
      "mean field equation",
      "parameters",
      "existence result",
      "exponential nonlinearities",
      "leray-schauder topological degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160203354J"
    }
  }
}