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 + ϕ u − 1 2 u △ u 2 = ∣ u ∣ p − 1 u + h ( x ) , in R 3 , − △ ϕ = u 2 , in R 3 , where 4 ≤ p + 1 < 12 , h ∈ L 2 ( R 3 ) 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).