On a class of nonlinear Schrödinger-Poisson systems involving a nonradial charge density

In the spirit of the classical work of P.H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger-Poisson system {equation presented} under different assumptions on ρ: R3→ R+at infinity. Our results cover the range p ∈ (2, 3) where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a 'limiting problem' at infinity and of the possible unboundedness of the Palais-Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem {equation presented}, in various functional settings which are suitable for both variational and perturbation methods.