2-peaks cluster solutions to the nonlinear Schrödinger-Bopp-Podolsky system

Gustavo de Paula Ramos

Published 2023-11-16Version 1

Suppose that $z_0$ is a local strict minimum point of $V \colon \mathbb{R}^3 \to ]0, \infty[$ and $V$ is adequately flat around $z_0$. We employ Lyapunov-Schmidt reduction to prove that if $\epsilon > 0$ is sufficiently small, then the nonlinear Schr\"odinger-Bopp-Podolsky system \[ \begin{cases} -\epsilon^2 \Delta u + (V + \phi) u = u |u|^{p-1}; \newline \Delta^2 \phi - \Delta \phi = 4 \pi u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] has a $2$-peaks solution and the corresponding peaks converge to $z_0$ as $\epsilon \to 0^+$, where $1 < p < 5$ and our unknowns are $u, \phi\colon\mathbb{R}^3\to\mathbb{R}$.