Existence results for non-local elliptic systems with Hardy-Littlewood-Sobolev critical nonlinearities

In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left{\begin{aligned} (-\Delta)^s u & = au+bv+\frac{2p}{p+q}\int_{\Omega}\frac{|v(y)|^q}{|x-y|^\mu}dy|u|^{p-2}u+2\xi_1\int_{\Omega}\frac{|u(y)|^{2^_\mu}}{|x-y|^\mu}dy|u|^{2^\mu-2}u,&& $\text{in }$ \Omega;\ (-\Delta)^s v & = bu+cv+\frac{2q}{p+q}\int{\Omega}\frac{|u(y)|^p}{|x-y|^\mu}dy|v|^{q-2}v+2\xi_2\int_{\Omega}\frac{|v(y)|^{2^_\mu}}{|x-y|^\mu}dy|v|^{2^_\mu-2}v,&& $\text{in }$ \Omega;\ u &=v=0,$\text{ in }$ \R^N\setminus\Omega, \end{aligned}\right. \end{equation*} where $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N>2s$, $s\in (0,1)$, ${\xi }_1,{\xi }_2\ge 0$, $(-\mathrm{\Delta }{)}^s$ is the well known fractional Laplacian, $\mu \in (0,N)$, $1<p,q\le 2_{\mu }^{\ast }$ where $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different parameters $p,q,{\xi }_1,$ and ${\xi }_2$, we are able to prove some existence and multiplicity results for the above equation by variational methods.