On the number of positive solutions of nonlinear systems

We prove that appropriate combinations of superlinearity and sublinearity of f(u) with respect to at zero and infinity guarantee the existence, multiplicity, and nonexistence of positive solutions to boundary value problems for the n-dimensional system ( (u )) + λh(t )f(u) = 0, 0 < t <1. The vector-valued function is defined by (u ) = (ϕ(u 1), . . . , ϕ(u n)), where u = (u1, . . . , un) and ϕ covers the two important cases ϕ(u ) = u and ϕ(u ) = |u |p−2u ,p >1. Our methods employ fixed point theorems in a cone.  2003 Elsevier Science (USA). All rights reserved.