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In the spirit of the classical work of P. H. Rabinowitz on nonlinearSchr\"odinger equations, we prove existence of mountain-pass solutions andleast energy solutions to the nonlinear Schr\"odinger-Poisson system\begin{equation}\nonumber \left\{\begin{array}{lll}  - \Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, \qquad &x\in \mathbb R^3,  \,\,\, -\Delta \phi=\rho(x) u^2,\ & x\in \mathbb R^3, \end{array} \right.\end{equation} under different assumptions on $\rho: \mathbb R^3\rightarrow\mathbb R_+$ at infinity. Our results cover the range $p\in(2,3)$ where thelack of compactness phenomena may be due to the combined effect of theinvariance by translations of a `limiting problem' at infinity and of thepossible unboundedness of the Palais-Smale sequences. Moreover, we findnecessary conditions for concentration at points to occur for solutions to thesingularly perturbed problem \begin{equation}\nonumber\left\{\begin{array}{lll}  - \epsilon^2\Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, \qquad &x\in \mathbbR^3,  \,\,\, -\Delta \phi=\rho(x) u^2,\ & x\in \mathbb R^3, \end{array} \right.\end{equation} in various functional settings which are suitable for bothvariational and perturbation methods.

On a class of nonlinear Schr\"odinger-Poisson systems involving a nonradial charge density