Abstract: In this paper, the following concave-convex fractional Schrödinger-Poisson system with doubly critical exponents is investigated
$\left\{\begin{array}{lr}(-\Delta)^s \boldsymbol{u}-\boldsymbol{\phi}|\boldsymbol{u}|^{2_s^*-3} \boldsymbol{u}=|\boldsymbol{u}|^{2_s^*-2} \boldsymbol{u}+\lambda h(\boldsymbol{x})|\boldsymbol{u}|^{q-2} \boldsymbol{u}, & \boldsymbol{x} \in \mathbf{R}^3, \\ (-\Delta)^s \boldsymbol{\phi}=|\boldsymbol{u}|^{2_s^*-1}, & \boldsymbol{x} \in \mathbf{R}^3,\end{array}\right.$
where 10 is a real parameter and h satisfies some certain conditions. It is showed that there exists λ^*>0 such that the system has a positive local minima solution with negative energy and a positive mountain-pass solution with positive energy for any λ∈(0,λ^{*) in Ds,2(R3) by applying the Mountain Pass Theorem and variational method.}

Key words: fractional Schrödinger-Poisson system, positive solutions, mountain pass theorem, doubly critical exponents