  We analyse the stability of inhomogeneous superfluid phases in a system
consisting of identical fermions confined in two layers that are immersed in a
Bose-Einstein condensate (BEC). The fermions in the two layers interact via an
induced interaction mediated by the BEC, which gives rise to pairing. We
present zero temperature phase diagrams varying the chemical potential
difference between the two layers and the range of the induced interaction, and
show that there is a large region where an inhomogeneous superfluid phase is
the ground state. This region grows with increasing range of the induced
interaction and it can be much larger than for a corresponding system with a
short range interaction. The range of the interaction is controlled by the
healing length of the BEC, which makes the present system a powerful tunable
platform to stabilise inhomogeneous superfluid phases. We furthermore analyse
the melting of the superfluid phases in the layers via phase fluctuations as
described by the Berezinskii-Kosterlitz-Thouless mechanism and show that the
normal, homogeneous and inhomogeneous superfluid phases meet in a tricritical
point. The superfluid density of the inhomogeneous superfluid phase is reduced
by inherent gapless excitations, and we demonstrate that this leads to a
significant suppression of the critical temperature as compared to the
homogeneous superfluid phase.
