Degenerate Fractional Kirchhoff-Type System with Magnetic Fields and Upper Critical Growth

Abstract

This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth:

$$\begin{aligned} \left\{ \begin{array}{lll} -\mathfrak {M}(\Vert u\Vert _{s,A}^2)[(-\Delta )^s_Au+u] = G_u(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal {I}_\mu *|u|^{p^*}\right) |u|^{p^*-2}u \ &{}\text{ in }\,\,\mathbb {R}^N,\\ \mathfrak {M}(\Vert v\Vert _{s,A})[(-\Delta )^s_Av+v] = G_v(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal {I}_\mu *|v|^{p^*}\right) |v|^{p^*-2}v \ &{}\text{ in }\,\,\mathbb {R}^N, \end{array}\right. \end{aligned}$$

where

$$\begin{aligned}\Vert u\Vert _{s,A}=\left( \iint _{\mathbb {R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}{\text {d}}x {\text {d}}y+\int _{\mathbb {R}^N}|u|^2{\text {d}}x\right) ^{1/2},\end{aligned}$$

and $(-\Delta )_{A}^s$ and A are called magnetic operator and magnetic potential, respectively, $\mathfrak {M}:\mathbb {R}^{+}_{0}\rightarrow \mathbb {R}^{+}_0$ is a continuous Kirchhoff function, $\mathcal {I}_\mu (x) = |x|^{N-\mu }$ with $0<\mu <N$, $C^1$-function G satisfies some suitable conditions, and $p^* =\frac{N+\mu }{N-2s}$. We prove the multiplicity results for this problem using the limit index theory. The novelty of our work is the appearance of convolution terms and critical nonlinearities. To overcome the difficulty caused by degenerate Kirchhoff function and critical nonlinearity, we introduce several analytical tools and the fractional version concentration-compactness principles which are useful tools for proving the compactness condition.