{
  "id": "1807.06114",
  "version": "v1",
  "published": "2018-07-16T21:13:44.000Z",
  "updated": "2018-07-16T21:13:44.000Z",
  "title": "Low energy nodal solutions to the Yamabe equation",
  "authors": [
    "Juan Carlos Fernández",
    "Jimmy Petean"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Given an isoparametric function $f$ on the $n$-dimensional sphere, we consider the space of functions $w\\circ f$ to reduce the Yamabe equation on the round sphere into a singular ODE on $w$ in the interval $[0,\\pi]$, of the form $w\" + (h(r)/\\sin r)w'+\\lambda(\\vert w\\vert^{4/n-2}w - w)=0$, where $h$ is a monotone function with exactly one zero on $[0,\\pi]$ and $\\lambda>0$ is a constant. The natural boundary conditions in order to obtain smooth solutions are $w'(0)=0$ and $w'(\\pi )=0$. We show that for any positive integer $k$ there exists a solution with exactly $k$-zeroes yielding solutions to the Yamabe equation with exactly $k$ connected isoparametric hypersurfaces as nodal set. The idea of the proof is to consider the initial value problems on both singularities $0$ and $\\pi$, and then to solve the corresponding double shooting problem, matching the values of $w$ and $w'$ at the unique zero of $h$. In particular we obtain solutions with exactly one zero, providing solutions of the Yamabe equation with low energy, which can be computed easily by numerical methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-16T21:13:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "low energy nodal solutions",
      "yamabe equation",
      "initial value problems",
      "natural boundary conditions",
      "isoparametric function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}