{
  "id": "2211.14895",
  "version": "v1",
  "published": "2022-11-27T17:28:29.000Z",
  "updated": "2022-11-27T17:28:29.000Z",
  "title": "Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity",
  "authors": [
    "Shiwang Ma",
    "Vitaly Moroz"
  ],
  "comment": "40 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\\Big(a+b\\int_{\\mathbb R^N}|\\nabla u|^2\\Big)\\Delta u+ \\lambda u= u^{q-1}+ u^{p-1} \\quad {\\rm in} \\ \\mathbb R^N, $$ as $\\lambda\\to 0$ and $\\lambda\\to +\\infty$, where $N=3$ or $N= 4$, $2<q\\le p\\le 2^*$, $2^*=\\frac{2N}{N-2}$ is the Sobolev critical exponent, $a>0$, $b\\ge 0$ are constants and $\\lambda>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $\\lambda\\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-\\Delta u+u=u^{q-1}$ and as $\\lambda\\to +\\infty$, after another rescaling the ground state solutions of the problem converge to a particular solution of the critical Emden-Fowler equation $-\\Delta u=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such rescalings, which depends in a non-trivial way on the space dimension $N=3$ and $N= 4$. We also discuss a connection of our results with a mass constrained problem associated to the Kirchhoff equation with the mass normalization constraint $\\int_{\\mathbb R^N}|u|^2=c^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-27T17:28:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35B25",
      "35B40"
    ],
    "keywords": [
      "nonlinear kirchhoff equation",
      "asymptotic profiles",
      "powers nonlinearity",
      "problem converge",
      "positive ground state solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}