Groundstates and infinitely many high energy solutions to a class of nonlinear Schr\"odinger-Poisson systems

We study a nonlinear Schr"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation $$-\mathrm{\Delta }u+u+{\lambda }^2(\frac{1}{\omega \mathrm{\mid }x{\mathrm{\mid }}^{N-2}}\star \rho u^2)\rho (x)u=\mathrm{\mid }u{\mathrm{\mid }}^{q-1}ux\in {\mathbb{R}}^N,$$ where $\omega =(N-2)\mathrm{\mid }{\mathbb{S}}^{N-1}\mathrm{\mid },$ $\lambda >0,$ $q\in (2,2^{\ast }-1),$ $\rho :{\mathbb{R}}^N\to \mathbb{R}$ is nonnegative and locally bounded, $N=3,4,5$ and $2^{\ast }=2N\mathrm{/}(N-2)$ is the critical Sobolev exponent. We prove existence and multiplicity of solutions working on a suitable finite energy space and under two separate assumptions which are compatible with instances where loss of compactness phenomena may occur.