  Exciton instability in graphene bilayer systems is studied in the case of a
short-ranged Coulomb interaction and a finite voltage difference between the
layers. Self-consistent exciton gap equations are derived and solved
numerically and analytically under controlled approximation. We obtain that a
critical strength of the Coulomb interaction exists for the formation of
excitons. The critical strength depends on the amount of voltage difference
between the layers and on the inter-layer hopping parameter.
