{
  "id": "1510.07231",
  "version": "v1",
  "published": "2015-10-25T10:37:11.000Z",
  "updated": "2015-10-25T10:37:11.000Z",
  "title": "An autonomous Kirchhoff-type equation with general nonlinearity in $\\mathbb{R}^N$",
  "authors": [
    "Sheng-Sen Lu"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the following autonomous Kirchhoff-type equation $$-\\left(a+b\\int_{\\mathbb{R}^N}|\\nabla{u}|^2\\right)\\Delta{u}= f(u),~~~~u\\in H^1(\\mathbb{R}^N),$$ where $a\\geq0,b>0$ are constants and $N\\geq1$. Under general assumptions on the nonlinearity $f$, we establish the existence results of a ground state and multiple solutions for $N\\geq2$, and obtain a nontrivial solution and its uniqueness, up to a translation and up to a sign, for $N=1$. The proofs are mainly based on a rescaling argument and a new description of the critical values in association with the level argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-25T10:37:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "35J60"
    ],
    "keywords": [
      "autonomous kirchhoff-type equation",
      "general nonlinearity",
      "general assumptions",
      "nontrivial solution",
      "existence results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151007231L"
    }
  }
}