  In this paper, we study a class of Schr\"{o}dinger-Poisson (SP) systems with
general nonlinearity where the nonlinearity does not require
Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new
estimates and explore the associated energy functional which is coercive and
bounded below on Sobolev space. Together with Ekeland variational principle, we
prove the existence of ground state solutions. Furthermore, when the `charge'
function is greater than a fixed positive number, the (SP) system possesses
only zero solutions. In particular, when `charge' function is radially
symmetric, we establish the existence of three positive solutions and the
symmetry breaking of ground state solutions.
