Concentrated solutions to the Schrödinger--Bopp--Podolsky system with a positive potential

Consider the Schrödinger–Bopp–Podolsky system $$\{\begin{array}{l}-{\epsilon }^2\mathrm{\Delta }u+(V+K\phi )u=u\mathrm{\mid }u{\mathrm{\mid }}^{p-1};\\ {\mathrm{\Delta }}^2\phi -\mathrm{\Delta }\phi =4\pi K{u}^2\end{array}\text{ in }{\mathbb{R}}^3$$ for sufficiently small $\epsilon >0$, where $V,K\text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\to [0,\mathrm{\infty }[$; $p\in ]1,5[$ are fixed and we want to solve for $u,\phi \text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\to \mathbb{R}$. Under certain hypotheses, we estimate the multiplicity of solutions in function of a critical manifold of $V$ and we establish the existence of solutions concentrated around critical points of $V$.