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 $lambda in (0,lambda_*)$ there are no positive solutions. Also, for $lambda ge lambda_*$ we show that the problem has a smallest positive solution $ar u_{lambda}$ and we investigate the continuity and monotonicity properties of the map $lambda o ar u_{lambda}$.