Nonlinear, nonhomogeneous parametric Neumann problems

Abstract

We consider a parametric nonlinear Neumann problem driven by a nonlinear nonhomogeneous differential operator, with a Carathéodory reaction $f$ which is $p$-superlinear in the second variable, but not necessarily satisfying the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation type result describing the dependence of positive solutions on the parameter $\lambda> 0$, show the existence of a smallest positive solution $\overline{u}_{\lambda}$ and investigate properties of the map $\lambda\mapsto\overline{u}_{\lambda}$. Finally, we show the existence of nodal solutions.