{
  "id": "math/0511193",
  "version": "v1",
  "published": "2005-11-08T08:01:53.000Z",
  "updated": "2005-11-08T08:01:53.000Z",
  "title": "Nonlinear eigenvalue problems in Sobolev spaces with variable exponent",
  "authors": [
    "Teodora Liliana Dinu"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the boundary value problem $-{\\rm div}((|\\nabla u|^{p\\_1(x) -2}+|\\nabla u|^{p\\_2(x)-2})\\nabla u)=f(x,u)$ in $\\Omega$, $u=0$ on $\\partial\\Omega$, where $\\Omega$ is a smooth bounded domain in $\\RR^N$. We focus on the cases when $f\\_\\pm (x,u)=\\pm(-\\lambda|u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x):=\\max\\{p\\_1(x),p\\_2(x)\\} < q(x) < \\frac{N\\cdot m(x)}{N-m(x)}$ for any $x\\in\\bar\\Omega$. In the first case we show the existence of infinitely many weak solutions for any $\\lambda>0$. In the second case we prove that if $\\lambda$ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a $\\ZZ\\_2$-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-08T08:01:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D05",
      "35J60",
      "35J70",
      "58E05",
      "68T40",
      "76A02"
    ],
    "keywords": [
      "nonlinear eigenvalue problems",
      "variable exponent",
      "sobolev spaces",
      "adequate variational methods",
      "nontrivial weak solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11193L"
    }
  }
}