  We prove a multiplicity result for \begin{equation*}
  \begin{cases}
  -\varepsilon^{2}\Delta_g u+\omega u+q^{2}\phi u=|u|^{p-2}u\\[1mm]
  -\Delta_g \phi +a^{2}\Delta_g^{2} \phi + m^2 \phi =4\pi u^{2}
  \end{cases}
  \text{ in }M, \end{equation*} where $(M,g)$ is a smooth and compact
$3$-dimensional Riemannian manifold without boundary, $p\in(4,6)$, $a,m,q\neq
0$, $\varepsilon>0$ small enough. The proof of this result relies on
Lusternik-Schnirellman category. We also provide a profile description for low
energy solutions.
