Electron. J. Diff. Eqns., Vol. 2008(2008), No. 78, pp. 1-13.

Boundary eigencurve problems involving the p-Laplacian operator

Abdelouahed El Khalil, Mohammed Ouanan

Abstract:
In this paper, we show that for each $\lambda \in \mathbb{R}$, there is an increasing sequence of eigenvalues for the nonlinear boundary-value problem
$$\displaylines{ \Delta_pu=|u|^{p-2}u \quad \hbox{in } \Omega\cr |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda \rho(x)|u|^{p-2}u+\mu|u|^{p-2}u \quad \hbox{on } \partial \Omega\,; }$$
also we show that the first eigenvalue is simple and isolated. Some results about their variation, density, and continuous dependence on the parameter $\lambda$ are obtained.

Submitted September 16, 2007. Published May 27, 2008.
Math Subject Classifications: 35P30, 35J20, 35J60.
Key Words: p-Laplacian operator; nonlinear boundary conditions; principal eigencurve; Sobolev trace embedding.

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Abdelouahed El Khalil
Department of mathematics, College of Sciences
Al-Imam Muhammad Ibn Saud Islamic University
P. Box 90950, Riyadh 11623 Saudi Arabia
email: alakhalil@imamu.edu.sa   lkhlil@hotmail.com
Mohammed. Ouanan
University Moulay Ismail, Faculty of Sciences et Technology
Department of informatics
P.O. Box 509, Boutalamine, 52000 Errachidia, Morocco
email: m_ouanan@hotmail.com

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