  This article concerns about the existence and multiplicity of weak solutions
for the following nonlinear doubly nonlocal problem with critical nonlinearity
in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{
\begin{split} (-\Delta)^su &= \lambda |u|^{q-2}u +
\left(\int_{\Omega}\frac{|v(y)|^{2^*_\mu}}{|x-y|^\mu}~\mathrm{d}y\right)
|u|^{2^*_\mu-2}u\; \text{in}\; \Omega
  (-\Delta)^sv &= \delta |v|^{q-2}v +
\left(\int_{\Om}\frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}~\mathrm{d}y \right)
|v|^{2^*_\mu-2}v \; \text{in}\; \Omega
  u &=v=0\; \text{in}\; \mb R^n\setminus\Omega, \end{split} \right.
\end{equation*} where $\Omega$ is a smooth bounded domain in $\mb R^n$, $n
>2s$, $s \in (0,1)$, $(-\Delta)^s$ is the well known fractional Laplacian, $\mu
\in (0,n)$, $2^*_\mu = \displaystyle\frac{2n-\mu}{n-2s}$ is the upper critical
exponent in the Hardy-Littlewood-Sobolev inequality, $1<q<2$ and
$\lambda,\delta >0$ are real parameters. We study the fibering maps
corresponding to the functional associated with $(P_{\lambda,\delta})$ and show
that minimization over suitable subsets of Nehari manifold renders the
existence of atleast two non trivial solutions of $(P_{\la,\delta})$ for
suitable range of $\la$ and $\delta$.
