{
  "id": "2403.06784",
  "version": "v1",
  "published": "2024-03-11T14:58:02.000Z",
  "updated": "2024-03-11T14:58:02.000Z",
  "title": "Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains",
  "authors": [
    "Haiyun Deng",
    "Jingwen Ji",
    "Feida Jiang",
    "Jiabin Yin"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we provide an affirmative answer to the conjecture A for bounded simple rotationally symmetric domains $\\Omega\\subset \\mathbb{R}^n(n\\geq 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the symmetry of positive solutions for two kinds of semilinear elliptic equations. To do this, when $f(\\cdot,s)$ is strictly convex with respect to $s$, we show that the nonnegativity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of $u$. Moreover, we prove the uniqueness of critical points of a positive solution to semilinear elliptic equation $-\\triangle u=f(\\cdot,u)$ with zero Dirichlet boundary condition for simple rotationally symmetric domains in $\\mathbb{R}^n$ by continuity method and a variety of maximum principles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-11T14:58:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B38",
      "35J05",
      "35J25"
    ],
    "keywords": [
      "semilinear elliptic equation",
      "higher dimensional domains",
      "simple rotationally symmetric domains",
      "critical points",
      "uniqueness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}