{
  "id": "2203.15843",
  "version": "v1",
  "published": "2022-03-29T18:23:55.000Z",
  "updated": "2022-03-29T18:23:55.000Z",
  "title": "Uniqueness for the Nonlocal Liouville Equation in $\\mathbb{R}$",
  "authors": [
    "Maria Ahrend",
    "Enno Lenzmann"
  ],
  "comment": "17 pages. Comments are welcome",
  "categories": [
    "math.AP",
    "math.DG",
    "nlin.SI"
  ],
  "abstract": "We prove uniqueness of solutions for the nonlocal Liouville equation $$ (-\\Delta)^{1/2} w = K e^w \\quad \\mbox{in $\\mathbb{R}$} $$ with finite total $Q$-curvature $\\int_{\\mathbb{R}} K e^w \\, dx< +\\infty$. Here the prescribed $Q$-curvature function $K=K(|x|) > 0$ is assumed to be a positive, symmetric-decreasing function satisfying suitable regularity and decay bounds. In particular, we obtain uniqueness of solutions in the Gaussian case with $K(x) = \\exp(-x^2)$. Our uniqueness proof exploits a connection of the nonlocal Liouville equation to ground state solitons for Calogero--Moser derivative NLS, which is a completely integrable PDE recently studied by P.~G\\'erard and the second author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-29T18:23:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlocal liouville equation",
      "uniqueness proof exploits",
      "ground state solitons",
      "decay bounds",
      "symmetric-decreasing function satisfying suitable regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}