{
  "id": "2212.07955",
  "version": "v1",
  "published": "2022-12-15T16:40:48.000Z",
  "updated": "2022-12-15T16:40:48.000Z",
  "title": "Ground state solution of a Kirchhoff type equation with singular potentials",
  "authors": [
    "Thanh Viet Phan"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We study the existence and blow-up behavior of minimizers for $E(b)=\\inf\\Big\\{\\mathcal{E}_b(u) \\,|\\, u\\in H^1(R^2), \\|u\\|_{L^2}=1\\Big\\},$ here $\\mathcal{E}_b(u)$ is the Kirchhoff energy functional defined by $\\mathcal{E}_b(u)= \\int_{R^2} |\\nabla u|^2 dx+ b(\\int_{R^2} |\\nabla u|^2d x)^2+\\int_{R^2} V(x) |u(x)|^2 dx - \\frac{a}{2} \\int_{R^2} |u|^4 dx,$ where $a>0$ and $b>0$ are constants. When $V(x)= -|x|^{-p}$ with $0<p<2$, we prove that the problem has (at least) a minimizer that is non-negative and radially symmetric decreasing. For $a\\ge a^*$ (where $a^*$ is the optimal constant in the Gagliardo-Nirenberg inequality), we get the behavior of $E(b)$ when $b\\to 0^+$. Moreover, for the case $a=a^*$, we analyze the details of the behavior of the minimizers $u_b$ when $b\\to 0^+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-15T16:40:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q40",
      "46N50"
    ],
    "keywords": [
      "kirchhoff type equation",
      "ground state solution",
      "singular potentials",
      "kirchhoff energy functional",
      "blow-up behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}