{
  "id": "2211.12106",
  "version": "v1",
  "published": "2022-11-22T09:16:27.000Z",
  "updated": "2022-11-22T09:16:27.000Z",
  "title": "Non-uniqueness for the nonlocal Liouville equation in $\\mathbb{R}$ and applications",
  "authors": [
    "Luca Battaglia",
    "Matteo Cozzi",
    "Antonio J. Fernández",
    "Angela Pistoia"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct multiple solutions to the nonlocal Liouville equation \\begin{equation} \\label{eqk} \\tag{L} (-\\Delta)^{\\frac{1}{2}} u = K(x) e^u \\quad \\mbox{ in } \\mathbb{R}. \\end{equation} More precisely, for $K$ of the form $K(x) = 1+\\varepsilon \\kappa(x)$ with $\\varepsilon \\in (0,1)$ small and $\\kappa \\in C^{1,\\alpha}(\\mathbb{R}) \\cap L^{\\infty}(\\mathbb{R})$ for some $\\alpha > 0$, we prove existence of multiple solutions to \\eqref{eqk} bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-22T09:16:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R11",
      "35A02",
      "35C08",
      "30F45"
    ],
    "keywords": [
      "nonlocal liouville equation",
      "multiple ground state soliton solutions",
      "applications",
      "non-uniqueness",
      "construct multiple solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}