{
  "id": "2202.10895",
  "version": "v1",
  "published": "2022-02-22T13:49:01.000Z",
  "updated": "2022-02-22T13:49:01.000Z",
  "title": "Qualitative analysis on the critical points of the Robin function",
  "authors": [
    "Francesca Gladiali",
    "Massimo Grossi",
    "Peng Luo",
    "Shusen Yan"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega\\subset\\mathbb{R}^N$ be a smooth bounded domain with $N\\ge2$ and $\\Omega_\\epsilon=\\Omega\\backslash B(P,\\epsilon)$ where $B(P,\\epsilon)$ is the ball centered at $P\\in\\Omega$ and radius $\\epsilon$. In this paper, we establish the number, location and non-degeneracy of critical points of the Robin function in $\\Omega_\\epsilon$ for $\\epsilon$ small enough. We will show that the location of $P$ plays a crucial role on the existence and multiplicity of the critical points. The proof of our result is a consequence of delicate estimates on the Green function near to $\\partial B(P,\\epsilon)$. Some applications to compute the exact number of solutions of related well-studied nonlinear elliptic problems will be showed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-22T13:49:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical points",
      "robin function",
      "qualitative analysis",
      "related well-studied nonlinear elliptic problems",
      "smooth bounded domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}