Asymptotic behaviour of principal eigenvalues for a class of cooperative systems

This paper analyzes the asymptotic behaviour as lambda up arrow infinity of the principal eigenvalue of the cooperative operator [GRAPHICS] in a bounded smooth domain 2 of R N, AT > 1, under homogeneous Dirichlet boundary conditions on a Q, where a >= 0, d >= 0, and b(x) > 0, c(x) > 0, for all x is an element of Omega. Precisely, our main result establishes that if Int(a + d)(-1) (0) consists of two components, Q(0,1) and Omega(0,2), then [GRAPHICS] where, for any D subset of Omega and lambda is an element of R, sigma(1)[(lambda); D] stands for the principal eigenvalue of 2 (.) in D. Moreover, if we denote by (phi(lambda),psi(lambda)) the principal eigenfunction associated to sigma [(lambda); Omega], normalized so that = 1, and, for instance, [GRAPHICS] then the limit [GRAPHICS] is well defined in Ho-0(1)(Omega) x H-0(1) (Omega), (phi= Psi = 0 in Omega \ Omega(0,1) and (phi,Psi)|Omega(0,1) provides us with the principal eigenfunction of . This is a rather striking result, for as, according to it, the principal ei-enfunction must approximate zero as oo if a + d > 0, in spite of the cooperative structure of the operator. (C) 2007 Elsevier Inc. All rights reserved.