On the number of positive solutions of nonlinear systems

We prove that appropriate combinations of superlinearity and sublinearity of f(u) with respect to Φ at zero and infinity guarantee the existence, multiplicity, and nonexistence of positive solutions to boundary value problems for the n-dimensional system (Φ(u′) ′ + λh(t)f(u) = 0, 0 < t < 1. The vector-valued function Φ is defined by Φ(u′) = ( (u₁′ ,..., (u_n′)), where u = (u₁,...,u_n and covers the two important cases (u′) = u′ and (u′) = u′ ^p-2u′, p > 1. Our methods employ fixed point theorems in a cone.