  Ultracold fermionic gases in optical lattices give a great opportunity for
creating different types of novel states. One of them is phase separation
induced by a trapping potential between different types of superfluid phases.
The core-shell structures, occurring in systems with a trapping potential, are
a good example of such separations. The types and the sequences of phases which
emerge in such structures can depend on spin-imbalance, shape of the trap and
on-site interaction strength. In this work, we investigate the properties of
such structures within an attractive Fermi gas loaded in the optical lattice,
in the presence of the trapping potential and their relations to the phase
diagram of the homogeneous system. Moreover, we show how external and internal
parameters of the system and parameters of the trap influence their properties.
In particular, we show a possible occurrence of the core-shell structure in a
system with a harmonic trap, containing the BCS and FFLO states. Additionally,
we find a spatial separation of two superfuild states in the system, one in the
BCS limit as well as the other one in the tightly bound local pairs (BEC)
regime.
