  Strong electronic correlations related to a repulsive local interaction
suppress the electronic compressibility in a single-band model, and the
capacitance of a corresponding metallic film is directly related to its
electronic compressibility. Both statements may be altered significantly when
two extensions to the system are implemented which we investigate here: (i) we
introduce an attractive nearest-neighbor interaction $V$ as antagonist to the
repulsive on-site repulsion $U$, and (ii) we consider nano-structured
multilayers (heterostructures) assembled from two-dimensional layers of these
systems. We determine the respective total compressibility $\kappa$ and
capacitance $C$ of the heterostructures within a strong coupling evaluation,
which builds on a Kotliar-Ruckenstein slave-boson technique. Whereas the
capacitance $C(n)$ for electronic densities $n$ close to half-filling is
suppressed---illustrated by a correlation induced dip in $C(n)$---it may be
appreciably enhanced close to a van Hove singularity. Moreover, we show that
the capacitance may be a non-monotonic function of $U$ close to half-filling
for both attractive and repulsive $V$. The compressibility $\kappa$ can differ
from $C$ substantially, as $\kappa$ is very sensitive to internal electrostatic
energies which in turn depend on the specific set-up of the heterostructure. In
particular, we show that a capacitor with a polar dielectric has a smaller
electronic compressibility and is more stable against phase separation than a
standard non-polar capacitor with the same capacitance.
