Degenerate fractional Kirchhoff-type system with magnetic fields and upper critical growth

TL;DR

This paper establishes multiplicity results for a degenerate fractional Kirchhoff system with magnetic fields and critical growth, featuring convolution terms and employing advanced analytical tools and concentration-compactness principles.

Contribution

It introduces new analytical methods and fractional concentration-compactness principles to handle degenerate Kirchhoff functions and critical nonlinearities in magnetic fractional systems.

Findings

Proved multiplicity of solutions for the system.

Developed fractional concentration-compactness principles.

Addressed challenges from convolution terms and critical growth.

Abstract

This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: $$ \left\{ \begin{array}{lll} -\mathfrak{M}(\|u\|_{s,A}^2)[(-\Delta)^s_Au+u] = G_u(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_\mu*|u|^{p^*}\right)|u|^{p^*-2}u \ &\mbox{in}\,\,\mathbb{R}^N,\\ \mathfrak{M}(\|v\|_{s,A})[(-\Delta)^s_Av+v] = G_v(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_\mu*|v|^{p^*}\right)|v|^{p^*-2}v \ &\mbox{in}\,\,\mathbb{R}^N, \end{array}\right. $$ where $$\|u\|_{s,A}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dx dy+\int_{\mathbb{R}^N}|u|^2dx\right)^{1/2},$$ 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.