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