{
  "id": "1810.05696",
  "version": "v1",
  "published": "2018-10-12T19:36:25.000Z",
  "updated": "2018-10-12T19:36:25.000Z",
  "title": "The $\\infty$-eigenvalue problem with a sign-changing weight",
  "authors": [
    "Uriel Kaufmann",
    "Julio D. Rossi",
    "Joana Terra"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega\\subset\\mathbb{R}^{n}$ be a smooth bounded domain and $m\\in C(\\overline{\\Omega})$ be a sign-changing weight function. For $1<p<\\infty$, consider the eigenvalue problem $$ \\left\\{ \\begin{array} [c]{ll} -\\Delta_{p}u=\\lambda m(x)|u|^{p-2}u & \\text{in }\\Omega,\\\\ u=0 & \\text{on }\\partial\\Omega, \\end{array} \\right. $$ where $\\Delta_{p}u$ is the usual $p$-Laplacian. Our purpose in this article is to study the limit as $p\\rightarrow\\infty$ for the eigenvalues $\\lambda _{k,p}\\left( m\\right) $ of the aforementioned problem. In addition, we describe the limit of some normalized associated eigenfunctions when $k=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-12T19:36:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "35P30",
      "35J60"
    ],
    "keywords": [
      "eigenvalue problem",
      "smooth bounded domain",
      "sign-changing weight function",
      "aforementioned problem",
      "normalized associated eigenfunctions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}