Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential

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, ·) 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 λ∈ R which we specify exactly in terms of principal eigenvalue of the differential operator.