Cluster semiclassical states of the nonlinear Schr\"odinger-Bopp-Podolsky system

TL;DR

This paper constructs multi-peak solutions for a nonlinear Schr"odinger-Bopp-Podolsky system in three dimensions, showing that under certain conditions on the potential and parameters, solutions cluster at the vertices of a regular polygon.

Contribution

It introduces a Lyapunov-Schmidt reduction approach to establish the existence of multi-peak cluster solutions in the system.

Findings

Existence of multi-peak solutions with peaks at polygon vertices.

Solutions form a cluster around a local minimum of the potential.

Results hold for sufficiently small epsilon and flat potential regions.

Abstract

Consider the following nonlinear Schr\"odinger-Bopp-Podolsky system in $\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p-1}; \\ a^2 \Delta^2 \phi - \Delta \phi = 4 \pi u^2, \end{cases} $$ where $a, \varepsilon > 0$; $1 < p < 5$; $V \colon \mathbb{R}^3 \to ]0, \infty[$ and we want to solve for $u, \phi \colon \mathbb{R}^3 \to \mathbb{R}$. 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 $z_0$.