{
  "id": "1405.0621",
  "version": "v1",
  "published": "2014-05-03T21:16:57.000Z",
  "updated": "2014-05-03T21:16:57.000Z",
  "title": "On the existence threshold for positive solutions of p-laplacian equations with a concave-convex nonlinearity",
  "authors": [
    "Fernando Charro",
    "Enea Parini"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the following boundary value problem with a concave-convex nonlinearity: \\begin{equation*} \\left\\{ \\begin{array}{r c l l} -\\Delta_p u & = & \\Lambda\\,u^{q-1}+ u^{r-1} & \\textrm{in }\\Omega, \\\\ u & = & 0 & \\textrm{on }\\partial\\Omega. \\end{array}\\right. \\end{equation*} Here $\\Omega \\subset \\mathbb{R}^n$ is a bounded domain and $1<q<p<r<p^*$. It is well known that there exists a number $\\Lambda_{q,r}>0$ such that the problem admits at least two positive solutions for $0<\\Lambda<\\Lambda_{q,r}$, at least one positive solution for $\\Lambda=\\Lambda_{q,r}$, and no positive solution for $\\Lambda > \\Lambda_{q,r}$. We show that \\[ \\lim_{q \\to p} \\Lambda_{q,r} = \\lambda_1(p), \\] where $\\lambda_1(p)$ is the first eigenvalue of the p-laplacian. It is worth noticing that $\\lambda_1(p)$ is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case $q=p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-03T21:16:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J70"
    ],
    "keywords": [
      "positive solution",
      "concave-convex nonlinearity",
      "existence threshold",
      "p-laplacian equations",
      "boundary value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0621C"
    }
  }
}