Concave-Convex problems for the robin p-laplacian plus an indefinite potential

We consider nonlinear Robin problems driven by the p-Laplacian plus and indefinite potential. In the reaction, we have the competing effects of a parametric concave (that is, (p-1)-sublinear) term and of a convex (that is, (p 1)-superlinear) term which need not satisfy the Ambrosetti-Rabinowitz condition. We prove a "bifurcation-type" theorem describing in a precise way the dependence the dependence of the set of positive solutions on the parameter λ > 0. In addition, we show the existence of a smallest positive solution and determine the monotonicity and continuity properties of the map.