Infinitely many small energy solutions for the Schrödinger-Poisson equations with magnetic field

In this paper, we consider the following Schrödinger-Poisson equations with magnetic field

$$(-i\mathrm{\nabla }-A(x){)}^{2}u+\theta (|x{|}^{-1}\ast |u{|}^2)u=f(|u{|}^2)u,u\in H^1({\mathbb{R}}^3,\mathbb{C}),$$

where $i$ is the imaginary unit and $\theta \ge 0$. The function $A:{\mathbb{R}}^3\to {\mathbb{R}}^3$ denotes a magnetic potential, and $V:{\mathbb{R}}^3\to \mathbb{R}$ is a continuous potential. First, we establish the existence of ground state solutions without imposing the strict monotonicity condition and Ambrosetti-Rabinowitz condition. Then using the dual fountain theorem, we obtain the existence of infinitely many small energy solutions. Our results extend some recent work in the literature.