Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system

Consider the following nonlinear Schrödinger-Bopp-Podolsky system in ${\mathbb{R}}^3$: $$\{\begin{array}{l}-{\epsilon }^2\mathrm{\Delta }u+(V+\phi )u=u\mathrm{\mid }u{\mathrm{\mid }}^{p-1};\\ a^2{\mathrm{\Delta }}^2\phi -\mathrm{\Delta }\phi =4\pi u^2,\end{array}$$ where $a,\epsilon >0$; $1<p<5$; $V\text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\to ]0,\mathrm{\infty }[$ and we want to solve for $u,\phi \text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\to \mathbb{R}$. By means of Lyapunov-Schmidt reduction, we show that if $K\ge 2$, $z_0$ is a strict local minimum of $V$, $V$ is adequately flat in a neighborhood of $z_0$ and $\epsilon$ is sufficiently small, then the system has a multipeak cluster solution with $K$ peaks placed at the vertices of a regular convex $K$-gon centered at $z_0$.