{
  "id": "1706.03034",
  "version": "v1",
  "published": "2017-06-09T16:49:33.000Z",
  "updated": "2017-06-09T16:49:33.000Z",
  "title": "Remarks on minimizers for $(p,q)$-Laplace equations with two parameters",
  "authors": [
    "Vladimir Bobkov",
    "Mieko Tanaka"
  ],
  "comment": "33 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the $(p,q)$-Laplace equation $-\\Delta_p u -\\Delta_q u = \\alpha |u|^{p-2}u + \\beta |u|^{q-2}u$ in a bounded domain $\\Omega \\subset \\mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $\\alpha, \\beta \\in \\mathbb{R}$. A curve on the $(\\alpha,\\beta)$-plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the $p$- and $q$-Laplacians under zero Dirichlet boundary condition are linearly independent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-09T16:49:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J62",
      "35J20",
      "35P30"
    ],
    "keywords": [
      "laplace equation",
      "zero dirichlet boundary condition",
      "parameters",
      "ground states",
      "corresponding energy levels"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}