{
  "id": "1904.03905",
  "version": "v1",
  "published": "2019-04-08T09:30:43.000Z",
  "updated": "2019-04-08T09:30:43.000Z",
  "title": "A monotonicity result under symmetry and Morse index constraints in the plane",
  "authors": [
    "Francesca Gladiali"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with solutions of semilinear elliptic equations of the type \\[ \\left\\{\\begin{array}{ll} -\\Delta u = f(|x|, u) \\qquad & \\text{ in } \\Omega, \\\\ u= 0 & \\text{ on } \\partial \\Omega, \\end{array} \\right. \\] where $\\Omega$ is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions $u$ that are invariant by rotations of a certain angle $\\theta$ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or $u$ is radial, or, else, there exists a direction $e\\in \\mathcal S$ such that $u$ is symmetric with respect to $e$ and it is strictly monotone in the angular variable in a sector of angle $\\frac{\\theta}2$. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-08T09:30:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "morse index constraints",
      "monotonicity result",
      "functions invariant",
      "semilinear elliptic equations",
      "nodal least-energy solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}