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

Abdelouahed El Khalil¹ ; Mohammed Ouanan²

¹ Département de Mathématiques & Génie Industriel École Polytechnique, Montréal Montréal (QC) H3C 3A7 Canada
² Departement of Mathematics Faculty of Sciences Dhar-Mahraz P.O. Box 1796 Atlas, Fez 30000, Morocco

Applicationes Mathematicae, Tome 32 (2005) no. 1, pp. 1-16.

Résumé

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 $\lambda$.

Détail

DOI : 10.4064/am32-1-1

Keywords: prove lambda bbb there increasing sequence eigenvalues lambda nonlinear boundary value problem cases mit delta p textrm mit omega nabla p partial partial lambda varrho p p textrm partial mit omega first lambda simple isolated prove results about variations density varrho continuity respect parameter lambda