{
  "id": "1606.07734",
  "version": "v1",
  "published": "2016-06-24T15:53:46.000Z",
  "updated": "2016-06-24T15:53:46.000Z",
  "title": "Explicit solutions and multiplicity results for some equations with the $p$-Laplacian",
  "authors": [
    "Philip Korman"
  ],
  "comment": "16 pages, 2 figures",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We derive explicit ground state solutions for several equations with the $p$-Laplacian in $R^n$, including (here $\\varphi (z)=z|z|^{p-2}$, with $p>1$) \\[ \\varphi \\left(u'(r)\\right)' +\\frac{n-1}{r} \\varphi \\left(u'(r)\\right)+u^M+u^Q=0 \\,. \\] The constant $M>0$ is assumed to be below the critical power, while $Q=\\frac{M p-p+1}{p-1}$ is above the critical power. This explicit solution is used to give a multiplicity result, similarly to C.S. Lin and W.-M. Ni [11]. We also give the $p$-Laplace version of G. Bratu's solution [3]. In another direction, we present a change of variables which removes the non-autonomous term $r^{\\alpha}$ in \\[ \\varphi \\left(u'(r)\\right)' +\\frac{n-1}{r} \\varphi \\left(u'(r)\\right)+r^{\\alpha} f(u)=0 \\,, \\] while preserving the form of this equation. In particular, we study singular equations, when $\\alpha <0$. The Coulomb case $\\alpha=-1$ turned out to give the critical power.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-24T15:53:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "35J61"
    ],
    "keywords": [
      "multiplicity result",
      "explicit solution",
      "critical power",
      "derive explicit ground state solutions",
      "study singular equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}