  As a potential candidate for the late-time accelerating expansion of the
Universe, the Chaplygin gas and its generalized models have significant
implications to modern cosmology. In this work we investigate the effects of
dark energy on the internal structure of a neutron star composed of two phases,
which leads us to wonder: Do stable neutron stars have a dark-energy core? To
address this question, we focus on the radial stability of stellar
configurations composed by a dark-energy core -- described by a Chaplygin-type
equation of state (EoS) -- and an ordinary-matter external layer which is
described by a polytropic EoS. We examine the impact of the rate of energy
densities at the phase-splitting surface, defined as $\alpha= \rho_{\rm
dis}^-/\rho_{\rm dis}^+$, on the radius, total gravitational mass and
oscillation spectrum. The resulting mass-radius diagrams are notably different
from dark energy stars without a common-matter crust. Specifically, it is found
that both the mass and the radius of the maximum-mass configuration decrease as
$\alpha$ becomes smaller. Furthermore, our theoretical predictions for
mass-radius relations consistently describe the observational measurements of
different massive millisecond pulsars as well as the central compact object
within the supernova remnant HESS J1731-347. The analysis of the normal
oscillation modes reveals that there are two regions of instability on the
$M(\rho_c)$ curve when $\alpha$ is small enough indicating that the usual
stability criterion $dM/d\rho_c>0$ still holds for rapid phase transitions.
However, this is no longer true for the case of slow transitions.
