On the principal eigencurve of the $p$-Laplacian related to the Sobolev trace embedding

Volume 32 / 2005

Abdelouahed El Khalil, Mohammed Ouanan Applicationes Mathematicae 32 (2005), 1-16 MSC: 35P30, 35J20, 35J60. DOI: 10.4064/am32-1-1

Abstract

We prove that for any $\lambda \in {\Bbb R}$, there is an increasing sequence of eigenvalues $\mu_n(\lambda)$ for the nonlinear boundary value problem $$ \cases{ {\mit\Delta}_pu=|u|^{p-2}u &\textrm{in } {\mit\Omega} ,\cr |\nabla u|^{p-2}{\partial u}/{\partial \nu}=\lambda \varrho (x)|u|^{p-2}u + \mu|u|^{p-2}u &\textrm{on } \partial {\mit\Omega} ,\cr} $$ and we show that the first one $\mu_{1}(\lambda)$ is simple and isolated; we also prove some results about variations of the density $\varrho $ and the continuity with respect to the parameter $\lambda$.

Authors

  • Abdelouahed El KhalilDépartement de Mathématiques & Génie Industriel
    École Polytechnique, Montréal
    Montréal (QC) H3C 3A7
    Canada
    e-mail
  • Mohammed OuananDepartement of Mathematics
    Faculty of Sciences Dhar-Mahraz
    P.O. Box 1796
    Atlas, Fez 30000, Morocco
    e-mail

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