Asymptotic behaviour of principal eigenvalues for a class of cooperative systems

This paper analyzes the asymptotic behaviour as λ↑∞ of the principal eigenvalue of the cooperative operator in a bounded smooth domain Ω of , N 1, under homogeneous Dirichlet boundary conditions on ∂Ω, where a 0, d 0, and b(x)>0, c(x)>0, for all . Precisely, our main result establishes that if Int(a+d)−1(0) consists of two components, Ω0,1 and Ω0,2, then where, for any D Ω and , stands for the principal eigenvalue of in D. Moreover, if we denote by (φλ,ψλ) the principal eigenfunction associated to , normalized so that , and, for instance, then the limit is well defined in , Φ=Ψ=0 in Ω Ω0,1 and (Φ,Ψ)Ω0,1 provides us with the principal eigenfunction of . This is a rather striking result, for as, according to it, the principal eigenfunction must approximate zero as λ↑∞ if a+d>0, in spite of the cooperative structure of the operator.