We study the existence, nonexistence and multiplicity of positive solutions for the family of problems - Delta u = f(lambda)(x, u), u is an element of H-0(1) (Omega), where Omega is a bounded domain in R-N, N >= 3 and lambda > 0 is a parameter. The results include the well-known nonlinearities of the Ambrosetti - Brezis - Cerami type in a more general form, namely lambda a( x) u(q) + b( x) u(p), where 0 <= q < 1 < p <= 2* - 1. The coefficient a( x) is assumed to be nonnegative but b( x) is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [ 9] are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods.82269286