Positive solutions for nonlinear nonhomogeneous parametric Robin problems

Abstract

Abstract We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter λ > 0 {\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 λ * {u^{*}_{\lambda}} of the problem, and we investigate the properties of the map λ ↦ u λ * {\lambda\mapsto u^{*}_{\lambda}} .