{
  "id": "1612.02102",
  "version": "v1",
  "published": "2016-12-07T02:46:11.000Z",
  "updated": "2016-12-07T02:46:11.000Z",
  "title": "Multiplicity of nodal solutions to the Yamabe problem",
  "authors": [
    "Mónica Clapp",
    "Juan Carlos Fernández"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\\geq 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-div_{g}(a\\nabla u)+bu=c|u|^{2^{\\ast}-2}u\\quad on\\ M$$, where $a,b,c\\in C^{\\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\\nabla)+b$ is coercive, and $2^{\\ast}=\\frac{2m}{m-2}$ is the critical Sobolev exponent. In particular, if $R_{g}$ denotes the scalar curvature of $(M,g)$, we give conditions which guarantee that the Yamabe problem $$\\Delta_{g}u+\\frac{m-2}{4(m-1} R_{g}u=\\kappa u^{2^{\\ast}-2}\\quad on\\ M$$ admits a prescribed number of nodal solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-07T02:46:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J61",
      "58J05",
      "35B06",
      "35B33",
      "35B44"
    ],
    "keywords": [
      "yamabe problem",
      "multiplicity",
      "compact riemannian manifold",
      "multiple nodal solutions",
      "symmetry assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}