Nehari Manifold for fractional Kirchhoff system with critical nonlinearity

In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\ \begin{equation} \left\{ \begin{array}{rllll} \mc L_M(u)&=\lambda f(x)|u|^{q-2}u+ \frac{2\alpha}{\alpha+\beta}\left|u\right|^{\alpha-2}u|v|^\beta & \text{in } \Omega,\\ \mc L_M(v)&=\mu g(x)|v|^{q-2}v+ \frac{2\beta}{\alpha+\beta}\left|u\right|^{\alpha}|v|^{\beta-2}v & \text{in } \Omega,\\ u&=v=0 &\mbox{in } \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{equation} where $\mc L_M(u)=M\left(\displaystyle \int_\Omega|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^{s} u $ is a double non-local operator due to Kirchhoff term $M(t)=a+b t$ with $a, b>0$ and fractional Laplacian $(-\Delta)^{s}, s\in(0, 1)$. We consider that $\Omega$ is a bounded domain in $\mathbb{R}^N$, {$2s<N\leq 4s$} with smooth boundary, $f, g$ are sign changing continuous functions, $\lambda, \mu>0$ are {real} parameters, $1<q<2$, $\alpha, \beta\ge 2$ {and} $\alpha+\beta=2_s^*={2N}/(N-2s)$ {is a fractional critical exponent}. Using the idea of Nehari manifold technique and a compactness result based on {classical idea of Brezis-Lieb Lemma}, we prove the existence of at least two positive solutions for $(\lambda, \mu)$ lying in a suitable subset of $\mathbb R^2_+$.