Fractional elliptic systems with critical nonlinearities

In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-\Delta)^s u = \frac{\alpha}{2_s^*}|u|^{\alpha-2}u|v|^{\beta}+f(x)\;\;\text{in}\;\mathbb{R}^{N}, v \frac{\beta}{2_s^*}|v|^{\beta-2}v|u|^{\alpha}+g(x)\;\;\text{in}\;\mathbb{R}^{N}, & \qquad u, \, >0\, \mbox{ in }\,\mathbb{R}^{N}, \end{aligned} \right. \end{equation*} where $N>2s$, $\alpha,\,\beta>1$, $\alpha+\beta=2N/(N-2s)$, and $f,\, g$ are nonnegative functionals dual space $\dot{H}^s(\mathbb{R}^{N})$. When $f=0=g$, show that ground state solution above is {\it unique}. On other hand, when $f$ $g$ nontrivial with ker$(f)$=ker$(g)$, then establish existence at least two different provided $\|f\|_{(\dot{H}^s)'}$ $\|g\|_{(\dot{H}^s)'}$ small enough. Moreover, also provide a global compactness result, which gives complete description Palais-Smale sequences system.