Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the 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 defined by Φ(u) = (q(t)φ(p(t)u₁), , q(t)φ(p(t)u_n)), where u = (u₁, ..., u_n), and φ covers the two important cases φ (u) = u and φ (u) = |u| ^p-2u, p > 1, h(t) = diag[h₁(t), ..., h_n(t)] and f(u) = (f¹(u), ..., fⁿ (u)). Assume that f ⁱ and h_i are nonnegative continuous. For u = (u ₁, , u_n), let f₀ⁱ = lim _∥u∥→0 fⁱ(u)/φ(∥u∥),f _∞ⁱ = lim _{∥u∥→∞} f ⁱ(u)/φ(∥u∥), (i=1,...,n), f₀ = max{f ₀¹, , 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 f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.