Existence, nonexistence and multiplicity of positive solutions for nonlinear, nonhomogeneous Neumann problems

We consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator and a strictly (p - 1)-sublinear reaction term. We prove a bifurcation-type result establishing the existence of a critical parameter value lambda* > 0 such that for all lambda > lambda* the problem has at least two positive solutions, for lambda = lambda* it has at least one positive solution and for lambda is an element of (0,lambda*) there are no positive solutions. Also, for lambda = lambda* we show that the problem has a smallest positive solution (u) over bar (lambda) and we investigate the continuity and monotonicity properties of the map lambda -> (u) over bar (lambda).