  We study excitonic condensation in an electron-hole bilayer system with
unequal layer densities at zero temperature. Using mean-field theory we solve
the BCS gap equations numerically and investigate the effects of intra-layer
interactions. We analyze the stability of the Sarma phase with $\bk,-\bk$
pairing by calculating the superfluid mass density and also by checking the
compressibility matrix. We find that with bare Coulomb interactions the
superfluid density is always positive in the Sarma phase, due to a peculiar
momentum structure of the gap function originating from the singular behavior
of the Coulomb potential at zero momentum and the presence of a sharp Fermi
surface. Introducing a simple model for screening, we find that the superfluid
density becomes negative in some regions of the phase diagram, corresponding to
an instability towards a Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) type superfluid
phase. Thus, intra-layer interaction and screening together can lead to a rich
phase diagram in the BCS-BEC crossover regime in electron-hole bilayer systems.
