  The possibility of "orbitally selective Mott transitions" within a multiband
Hubbard model, in which one orbital with large on-site electron-electron
repulsion $U_1$ is insulating and another orbital, to which it is hybridized,
with small $U_{-1}$, is metallic, is a problem of long-standing debate and
investigation. In this paper we study an analogous phenomenon, the co-existence
of metallic and insulating bands in a system of orbitals with different
electron-phonon coupling (EPC). To this end, we examine two variants of the
bilayer Holstein model: a uniform bilayer and a "Holstein-Metal interface"
where the electron-phonon coupling, $\lambda$, is zero in the "metallic" layer.
In the uniform bilayer Holstein model, charge density wave (CDW) order
dominates at small interlayer hybridization $t_3$, but decreases and eventually
vanishes as $t_3$ grows, providing a charge analog of singlet (spin liquid)
physics. In the interface case, we show that CDW order penetrates into the
metal layer and forms long-range CDW order at intermediate ratio of inter- to
intra-layer hopping strengths, $1.4 \lesssim t_3/t \lesssim 3.4$. This is
consistent with the occurrence of an "orbitally selective CDW" regime at weak
$t_3$ in which the layer with $\lambda_{1} \neq 0$ exhibits long-range charge
order, but the "metallic layer" with $\lambda_{-1}=0$, to which it is
hybridized, does not.
