Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(uʹ))ʹ + 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 , 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