{
  "id": "1605.01875",
  "version": "v1",
  "published": "2016-05-06T09:49:09.000Z",
  "updated": "2016-05-06T09:49:09.000Z",
  "title": "Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results",
  "authors": [
    "Aleks Jevnikar",
    "Wen Yang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We are concerned with the following class of equations with exponential nonlinearities: $$ \\Delta u+h_1e^u-h_2e^{-2u}=0 \\qquad \\mbox{in } B_1\\subset\\mathbb{R}^2, $$ which is related to the Tzitz\\'eica equation. Here $h_1, h_2$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitz\\'eica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-06T09:49:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J61",
      "35J20",
      "35R01",
      "35B44"
    ],
    "keywords": [
      "tzitzéica equation",
      "blow-up analysis",
      "analytic aspects",
      "tzitzeica equation",
      "general existence result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}