Asymptotic behaviour of principal eigenvalues for a class of cooperative systems

详细信息    查看全文

• 作者：Pablo Á ; lvarez Caudevilla ; Juliá ; n Ló ; pez-Gó ; mez
• 关键词：Cooperative systems ; Principal eigenvalues ; Degenerate problem ; Asymptotic behaviour ; Lower estimates through the Lebesgue measure
• 刊名：Journal of Differential Equations
• 出版年：2008
• 出版时间：1 March 2008
• 年：2008
• 卷：244
• 期：5
• 页码：1093-1113
• 全文大小：202 K

文摘

This paper analyzes the asymptotic behaviour as λ↑∞ of the principal eigenvalue of the cooperative operator

in a bounded smooth domain Ω of , N1, under homogeneous Dirichlet boundary conditions on ∂Ω, where a0, d0, 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, Ω_0,1 and Ω_0,2, then

where, for any DΩ and , stands for the principal eigenvalue of in D. Moreover, if we denote by (φ_λ,ψ_λ) the principal eigenfunction associated to , normalized so that , and, for instance,

then the limit

is well defined in , Φ=Ψ=0 in ΩΩ_0,1 and (Φ,Ψ)|_Ω0,1 provides us with the principal eigenfunction of 10928cf777"">. 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.