{
  "id": "1411.5192",
  "version": "v1",
  "published": "2014-11-19T12:00:01.000Z",
  "updated": "2014-11-19T12:00:01.000Z",
  "title": "On positive solutions for $(p,q)$-Laplace equations with two parameters",
  "authors": [
    "Vladimir Bobkov",
    "Mieko Tanaka"
  ],
  "comment": "28 pages, 3 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-\\Delta_p u -\\Delta_q u = \\alpha |u|^{p-2} u + \\beta |u|^{q-2} u$, where $p \\neq q$, under the zero Dirichlet boundary condition in $\\Omega$. The main result of our research is the construction of a continuous curve in $(\\alpha,\\beta)$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $\\Omega$ for which the corresponding first Dirichlet eigenvalue of $-\\Delta_p$ is not monotone w.r.t. $p > 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-19T12:00:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J62",
      "35J20",
      "35P30"
    ],
    "keywords": [
      "positive solutions",
      "laplace equation",
      "zero dirichlet boundary condition",
      "parameters",
      "corresponding first dirichlet eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}