  The theoretical description of compact structures that share some key
features with mass varying particles allows for a simple analysis of
equilibrium and stability for massive stellar bodies. We investigate static,
spherically symmetric solutions of Einstein equations for a system composed by
nonbaryonic matter (neutrinos or dark matter) which forms stable structures
through attractive forces mediated by a background scalar-field (dark energy).
Assuming that the dark matter, or massive neutrinos, consist of a gas of weakly
interacting particles, the coupling with the scalar field is translated into an
effective dependence of the mass of the compounding particle on the radial
coordinate of the curved spacetime. The stability analysis reveals that these
static solutions become dynamically unstable for different Buchdahl limits of
the ratio between the total mass-energy and the stellar radius, $M/R$. We also
find regular solutions that for an external observer resemble Schwarzschild
black-holes. Our analysis leaves unanswered the question whether such
solutions, which are both regular and stable, do exist.
