International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 10, Pages 1555-1563
doi:10.1155/IJMMS.2005.1555

Simplicity and stability of the first eigenvalue of a nonlinear elliptic system

Abdelouahed El Khalil,1 Said El Manouni,2,3 and Mohammed Ouanan3

1Département de Mathématiques et Génie Industriel, Ecole Polytechnique de Montréal, H3C 3A7, Canada
223 Biesentaler Street, Berlin 13359, Germany
3Department of Mathematics, Faculty of Sciences Dhar-Mahraz, P.O. Box 1796 Atlas, Fez 30000, Morocco

Received 8 July 2004; Revised 17 March 2005

Copyright © 2005 Abdelouahed El Khalil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove some properties of the first eigenvalue for the elliptic system −Δpu=λ|u|α|v|βv in Ω, −Δqv=λ|u|α|v|βu in Ω, (u,v)∈W01,p(Ω)×W01,q(Ω). In particular, the first eigenvalue is shown to be simple. Moreover, the stability with respect to (p,q) is established.