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

Abstract

Consider the following nonlinear Schrödinger-Bopp-Podolsky system in $\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 Δu + (V + φ) u = u |u|^{p-1}; \\ a^2 Δ^2 φ- Δφ= 4 πu^2, \end{cases} $$ where $a, \varepsilon > 0$; $1 < p < 5$; $V \colon \mathbb{R}^3 \to ]0, \infty[$ and we want to solve for $u, φ\colon \mathbb{R}^3 \to \mathbb{R}$. By means of Lyapunov-Schmidt reduction, we show that if $K \geq 2$, $z_0$ is a strict local minimum of $V$, $V$ is adequately flat in a neighborhood of $z_0$ and $\varepsilon$ 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 $z_0$.