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 $M(t)=a+bt$ with $a,b>0$ and fractional Laplacian $(-\mathrm{\Delta }{)}^s,s\in (0,1)$. We consider that $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$, {$2s<N\le 4s$} with smooth boundary, $f, g$aresignchangingcontinuousfunctions,$\lambda, \mu>0$ are {real} parameters, $1<q<2$, $\alpha ,\beta \ge 2$ {and} $\alpha +\beta =2_s^{\ast }=2N\mathrm{/}(N-2s)$ {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 ${\mathbb{R}}_{+}^2$.