{
  "id": "1709.01106",
  "version": "v1",
  "published": "2017-09-04T18:20:28.000Z",
  "updated": "2017-09-04T18:20:28.000Z",
  "title": "Bubbling solutions for Moser-Trudinger type equations on compact Riemann surfaces",
  "authors": [
    "Pablo Figueroa",
    "Monica Musso"
  ],
  "comment": "41 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study an elliptic equation related to the Moser-Trudinger inequality on a compact Riemann surface $(S,g)$, $$ \\Delta_g u+\\lambda \\Biggl(ue^{u^2}-{1\\over |S|} \\int_S ue^{u^2} dv_g\\Biggl)=0,\\quad\\text{in $S$},\\qquad \\int_S u\\,dv_g=0, $$ where $\\lambda>0$ is a small parameter, $|S|$ is the area of $S$, $\\Delta_g$ is the Laplace-Beltrami operator and $dv_g$ is the area element. Given any integer $k\\geq 1$, under general conditions on $S$ we find a bubbling solution $u_\\lambda$ which blows up at exactly $k$ points in $S$, as $\\lambda \\to0$. When $S$ is a flat two-torus in rectangular form, we find that either seven or nine families of such solutions do exist for $k=2$. In particular, in any square flat two-torus actually nine families of bubbling solutions with two bubbling points do exist. If $S$ is a Riemann surface with non-constant Robin's function then at least two bubbling solutions with $k=1$ exists.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-04T18:20:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J08",
      "35J15",
      "53C20"
    ],
    "keywords": [
      "compact riemann surface",
      "bubbling solution",
      "moser-trudinger type equations",
      "square flat two-torus",
      "non-constant robins function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}