  This work theoretically analyzes electronic instabilities in AA-stacked
bilayer graphene and the role of the Coulomb interaction in stabilization of
electronic ordered states in this system. The Coulomb interaction screening is
accounted with the help of the random phase approximation. We observe that the
repulsion between charge carriers located in the same graphene layers is much
larger than that between the electrons moving in different layers (moreover,
under certain circumstances the inter-layer effective interaction becomes
attractive). At zero doping, the intra-layer Coulomb repulsion is responsible
for arising of the spin-density wave (SDW) state. The Neel temperature for this
phase is estimated to be about several tens K. The SDW is the dominating order
in undoped system, but it is destroyed completely by sufficiently strong
doping, allowing a superconducting phase to emerge. As for the superconducting
mechanism, we demonstrate that the effective inter-layer attraction can give
rise to superconductivity. However, the corresponding critical temperature is
negligibly small, and phonon-mediated attraction must be introduced into the
model to make the superconductivity observable. Strong intra-layer Coulomb
repulsion favors the order parameters that couple electrons in different
layers. In this case two different types of the superconducting phase can exist
in the system. The first type is a usual superconductivity with Copper pairs
having zero total momentum in the ground state. Such superconductivity exists
if the coupling between electrons with momenta near different Dirac cones
dominates. When the interaction between electrons near the same Dirac cones is
larger, the Cooper pairs have a non-zero momenta in the ground state.
Superconducting gap can increase or decrease with doping depending on the
relationship between intra-band and inter-band BCS interaction.
