Positive solutions for nonlinear Robin problems with indefinite potential and competing nonlinearities

Abstract

We consider a nonlinear Robin problem associated to the p-Laplacian plus an indefinite potential. In the reaction we have the competing effects of two nonlinear terms. One is parametric and strictly $(p-1)$-sublinear. The other is $(p-1)$-linear. We prove a bifurcation-type theorem describing the dependence of the set of positive solutions on the parameter $\lambda >0$. We also show that for every admissible parameter the problem has a smallest positive solution ${\bar{u}}_{\lambda }$ and we study monotonicity and continuity properties of the map $\lambda \rightarrow {\bar{u}}_{\lambda }$.