Positive solutions for nonlinear nonhomogeneous parametric Robin problems

Abstract

We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carath\'eodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter $\lambda>0$ approaches zero we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally we show that for every admissible parameter value there is a smallest positive solution $u^*_{\lambda}$ of the problem and we investigate the properties of the map $\lambda\mapsto u^*_{\lambda}$.