We consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation f(z,cdot ) is (p-1)-sublinear and then the case where it is (p-1)-superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter lambda in {mathbb {R}} which we specify exactly in terms of principal eigenvalue of the differential operator.