Ground state of a magnetic nonlinear Choquard equation

We consider the stationary magnetic nonlinear Choquard equation $$-(\mathrm{\nabla }+iA(x){)}^{2}u+V(x)u=(\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}\ast F(\mathrm{\mid }u\mathrm{\mid }))\frac{f(\mathrm{\mid }u\mathrm{\mid })}{\mathrm{\mid }u\mathrm{\mid }}u,$$ where $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a vector potential, $V$ is a scalar potential, $f\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ and $F$ is the primitive of $f$. Under mild hypotheses, we prove the existence of a ground state solution for this problem. We also prove a simple multiplicity result by applying Ljusternik-Schnirelmann methods.