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

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1. Verfasser: Ramos, Gustavo de Paula
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Veröffentlicht: 2023
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author Ramos, Gustavo de Paula
author_facet Ramos, Gustavo de Paula
contents 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 $z_0$.
format Preprint
id arxiv_https___arxiv_org_abs_2311_09949
institution arXiv
publishDate 2023
record_format arxiv
spellingShingle Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system
Ramos, Gustavo de Paula
Analysis of PDEs
35J48, 35B40, 35Q40
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 $z_0$.
title Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system
topic Analysis of PDEs
35J48, 35B40, 35Q40
url https://arxiv.org/abs/2311.09949