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 systemunder different assumptions on ρ : R 3 → 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 problemin various functional settings which are suitable for both variational and perturbation methods. MSC: 35J20, 35B65, 35J60, 35Q55

We study a nonlinear Schrödinger-Poisson system which reduces to the nonlinear and nonlocal PDEis nonnegative, locally bounded, and possibly non-radial, N = 3, 4, 5 and 2 * = 2N/(N − 2) is the critical Sobolev exponent. In our setting ρ is allowed as particular scenarios, to either 1) vanish on a region and be finite at infinity, or 2) be large at infinity. We find least energy solutions in both cases, studying the vanishing case by means of a priori integral bounds on the Palais-Smale sequences and highlighting the role of certain positive universal constants for these bounds to hold. Within the Ljusternik-Schnirelman theory we show the existence of infinitely many distinct pairs of high energy solutions, having a min-max characterisation given by means of the Krasnoselskii genus. Our results cover a range of cases where major loss of compactness phenomena may occur, due to the possible unboundedness of the Palais-Smale sequences, and to the action of the group of translations. MSC: 35Q55,

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