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

We prove a multiplicity result for $$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^{2}\Delta _g u+\omega u+q^{2}\phi u=|u|^{p-2}u\\ -\Delta _g \phi +a^{2}\Delta _g^{2} \phi + m^2 \phi =4\pi u^{2} \end{array}\right. } \text { in }M, \end{aligned}$$ where (M, g) is a smooth and compact 3-dimensional Riemannian manifold without boundary, $$p\in (4,6)$$ , $$a,m,q\ne 0$$ , $$\varepsilon >0$$ small enough. The proof of this result relies on Lusternik–Schnirellman category. We also provide a profile description for low energy solutions.