An eigenvalue problem for quasilinear systems

The paper deals with the existence of positive solutions for the n-dimensional quasilinear system (Φ(u′))′ + λh(t)f (u) = 0, 0 < t < 1, with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is denned by Φ(u) = ((μ₁),..., (μ_n)), where u = (u₁,...,u_n), and ψ covers the two important cases ψ(u) = u and ψ(u) = |u|^p-2u, p > 1, h(t)=diag [h₁(t),...,h_n(t)] and f(u) = (f ¹(u),...,fⁿ(u)). Assume that fⁱ and h _i are nonnegative continuous. For u = (u₁,... ,u _n), let f₀ⁱ = lim_||u||→00 fⁱ(u)/ψ(||u||), f_∞ⁱ = lim _||u||→∞ fⁱ(u)ψ(||u||), i = 1,..., n, f ₀ = max{f₀¹,..., f₀ⁿ} and f_∞ = max{f_∞¹,... , f _∞ⁿ}. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f ₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed point theorems in a cone.