Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields

TL;DR

This paper investigates multiple solutions that concentrate around minima for a fractional Kirchhoff equation with magnetic fields, revealing how the solutions' multiplicity relates to the topology of the potential's minimum set.

Contribution

It introduces new multiplicity results for fractional Kirchhoff equations with magnetic fields using penalization and topological methods.

Findings

Multiple solutions concentrate near potential minima.

The number of solutions is related to the topology of the minimum set.

Solutions exhibit concentration behavior as the parameter ε approaches zero.

Abstract

This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential having a local minimum and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Applying penalization techniques and Ljusternik-Schnirelman theory, we relate the number of nontrivial solutions with the topology of the set where the potential $V$ attains its minimum.