Asymptotic behaviour of principal eigenvalues for a class of cooperative systems

This paper analyzes the asymptotic behaviour as λ↑∞λ↑∞ of the principal eigenvalue of the cooperative operatorL(λ):=(L1+λa(x)−b−cL2+λd(x)) in a bounded smooth domain Ω   of RNRN, N⩾1N⩾1, under homogeneous Dirichlet boundary conditions on ∂Ω  , where a⩾0a⩾0, d⩾0d⩾0, and b(x)>0b(x)>0, c(x)>0c(x)>0, for all x∈Ω¯. Precisely, our main result establishes that if Int(a+d)−1(0)Int(a+d)−1(0) consists of two components, Ω0,1Ω0,1 and Ω0,2Ω0,2, thenlimλ↑∞σ1[L(λ);Ω]=mini∈{1,2}σ1[L(0);Ω0,i], where, for any D⊂ΩD⊂Ω and λ∈Rλ∈R, σ1[L(λ);D]σ1[L(λ);D] stands for the principal eigenvalue of L(λ)L(λ) in D  . Moreover, if we denote by (φλ,ψλ)(φλ,ψλ) the principal eigenfunction associated to σ[L(λ);Ω]σ[L(λ);Ω], normalized so that ∫Ω(φλ2+ψλ2)=1, and, for instance,σ1[L(0);Ω0,1]<σ1[L(0);Ω0,2],σ1[L(0);Ω0,1]<σ1[L(0);Ω0,2], then the limit(Φ,Ψ):=limλ↑∞(φλ,ψλ) is well defined in H01(Ω)×H01(Ω), Φ=Ψ=0Φ=Ψ=0 in Ω∖Ω0,1Ω∖Ω0,1 and (Φ,Ψ)|Ω0,1(Φ,Ψ)|Ω0,1 provides us with the principal eigenfunction of σ[L(0);Ω0,1]σ[L(0);Ω0,1]. This is a rather striking result, for as, according to it, the principal eigenfunction must approximate zero as λ↑∞λ↑∞ if a+d>0a+d>0, in spite of the cooperative structure of the operator.