Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields

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)(-Δ)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\epsilon >0$ is a small parameter, $a,b>0$ are constants, $s\in (\frac{3}{4},1)$, $(-\Delta {)}_A^s$ is the fractional magnetic Laplacian, $A:{\mathbb{R}}^3\to {\mathbb{R}}^3$ is a smooth magnetic potential, $V:{\mathbb{R}}^3\to \mathbb{R}$ is a positive continuous potential having a local minimum and $f:\mathbb{R}\to \mathbb{R}$ 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.

arXiv admin note: text overlap with arXiv:1807.07444, arXiv:1808.01925