  A layered system of two-dimensional planes containing fermionic polar
molecules can potentially realize a number of exotic quantum many-body states.
Among the predictions, are density-wave instabilities driven by the anisotropic
part of the dipole-dipole interaction in a single layer. However, in typical
multilayer setups it is reasonable to expect that the onset and properties of a
density-wave are modified by adjacent layers. Here we show that this is indeed
the case. For multiple layers the critical strength for the density-wave
instability decreases with the number of layers. The effect depends on density
and is more pronounced in the low density regime. The lowest solution of the
instability corresponds to the density waves in the different layers being
in-phase, whereas higher solutions have one or several adjancet layers that are
out of phase. The parameter regime needed to explore this instability is within
reach of current experiments.
