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

Abstract: 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 $\Omega$ is a smooth bounded domain in $\R^N$, $N>2s$, $s\in(0,1)$, $\xi_1,\xi_2\geq 0$, $(-\Delta)^s$ is the well known fractional Laplacian, $\mu\in(0,N)$, $1<p,q\leq 2^*_\mu$ where $2^*_\mu=\frac{2N-\mu}{N-2s}$ 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.