{
  "id": "1907.09895",
  "version": "v1",
  "published": "2019-07-23T14:12:32.000Z",
  "updated": "2019-07-23T14:12:32.000Z",
  "title": "On the number of critical points of solutions of semilinear equations in $\\mathbb{R}^2$",
  "authors": [
    "Francesca Gladiali",
    "Massimo Grossi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we construct families of domains $\\Omega_\\varepsilon$ and solutions $u_\\varepsilon$ of \\[\\begin{cases} -\\Delta u_\\varepsilon=1&\\text{ in }\\ \\Omega_\\varepsilon\\\\ u_\\varepsilon=0&\\text{ on }\\ \\partial\\Omega_\\varepsilon \\end{cases}\\] such that, for any integer $k\\ge2$, $u_\\varepsilon$ admits at least $k$ maxima points. The domain $\\Omega_\\varepsilon$ is \"not far\" to be convex in the sense that it is starshaped, the curvature of $\\partial\\Omega_\\varepsilon$ changes sign $once$ and the minimum of the curvature of $\\partial\\Omega_\\varepsilon$ goes to $0$ as $\\varepsilon\\to0$. Extensions to more general nonlinear elliptic problems will be provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-23T14:12:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semilinear equations",
      "critical points",
      "general nonlinear elliptic problems",
      "changes sign",
      "construct families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}