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

This paper deals with the following degenerate fractional Kirchhoff-type system magnetic fields and critical growth: $$\begin{aligned} \left\{ \begin{array}{lll} -\mathfrak {M}(\Vert u\Vert _{s,A}^2)[(-\Delta )^s_Au+u] = G_u(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal {I}_\mu *|u|^{p^*}\right) |u|^{p^*-2}u \ &{}\text{ in }\,\,\mathbb {R}^N,\\ \mathfrak v\Vert _{s,A})[(-\Delta )^s_Av+v] G_v(|x|,|u|^2,|v|^2) *|v|^{p^*}\right) |v|^{p^*-2}v {R}^N, \end{array}\right. \end{aligned}$$ where $$\begin{aligned}\Vert _{s,A}=\left( \iint _{\mathbb {R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}{\text {d}}x {\text {d}}y+\int {R}^N}|u|^2{\text {d}}x\right) ^{1/2},\end{aligned}$$ $$(-\Delta )_{A}^s$$ A are called operator potential, respectively, $$\mathfrak {M}:\mathbb {R}^{+}_{0}\rightarrow \mathbb {R}^{+}_0$$ is a continuous Kirchhoff function, $$\mathcal (x) |x|^{N-\mu }$$ $$0<\mu <N$$ , $$C^1$$ -function G satisfies some suitable conditions, $$p^* =\frac{N+\mu }{N-2s}$$ . We prove multiplicity results for this problem using limit index theory. The novelty of our work appearance convolution terms nonlinearities. To overcome difficulty caused by function nonlinearity, we introduce several analytical tools version concentration-compactness principles which useful proving compactness condition.