Multiple solutions and profile description for a nonlinear Schrödinger-Bopp-Podolsky-Proca system on a manifold

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