Local “superlinearity” and “sublinearity” for the p-Laplacian

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems − pu = fλ(x, u), u ∈ W 1,p 0 (Ω), where Ω is a bounded domain in RN, N >p, and λ > 0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti–Brezis–Cerami type in a more general form, namely λa(x)uq + b(x)ur, where 0 q