Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger-Poisson systems

In the present paper, we study the following modified Schrödinger-Poisson system (P){−△u+V(x)u+ϕu−12u△u2=∣u∣p−1u+h(x),inR3,−△ϕ=u2,inR3, where 4≤p+1<12, h∈L2(R3) and h≠0. This model describes the interaction of a charge particle with an electro-magnetic field. Under some hypotheses on V(x), we prove the existence of a non-trivial ground state solution and two non-trivial ground state solutions for the system (P).