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

Abstract

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, $p\in(4,6)$, $a,m,q\neq 0$, $\varepsilon>0$ small enough. The proof of this result relies on Lusternik-Schnirellman category. We also provide a profile description for low energy solutions.