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)(p(t)u1), …, q(t)(p(t)un)), where u = (u1, …, un), andcovers the two important cases (u) = u and (u) = ‖u‖p > 1, h(t) = diag[h1(t), …, hn(t)] and f(u) = (f1(u), …, fn (u)). Assume that fi and hi are nonnegative continuous. For u = (u1, …, un), let f0i=lim⁡||u||→0fi(u)ϕ(||u||),f∞i=lim⁡||u||→∞fi(u)ϕ(||u||),(i=1,…,n), f0 = maxf10, …, fn0∼ and f∞ = maxf1∞, …, fn∞∼. 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.