  We have considered the AB-stacked bilayer graphene Josephson junction. The
bilayers are supposed to be in the charge equilibrium states and at the
half-filling in each of the electronic layers of the construction and at each
value of the external gate. By considering the interacting bilayers in both
sides of the junction and by taking into account both intralayer and interlayer
Coulomb interaction effects, we have calculated the normal and excitonic tunnel
currents through the junction. The electronic band renormalizations have been
taken into account, due to the excitonic pairing effects and condensation in
the BLGs. The exact four-band energy dispersions, including the excitonic
renormalizations, have been used for the bilayers without any low-energy
approximation. We show the degeneracy of the ground state at the zero applied
voltage and for different phases of the coherent condensates by showing a dc
Josephson current through the junction. The normal and excitonic tunneling
currents have been calculated for different gate voltages and for different
values of the interaction parameters. The role of the charge neutrality point
has been discussed in details.
