  We report about stability conditions for static, spherically symmetric
objects that share the essential features of mass varying neutrinos in
cosmological scenarios. Compact structures of particles with variable mass are
held together preponderantly by an attractive force mediated by a background
scalar field. Their corresponding conditions for equilibrium and stability are
given in terms of the ratio between the total mass-energy and the spherical
lump radius, $M/R$. We show that the mass varying mechanism leading to lump
formation can modify the cosmological predictions for the cosmological neutrino
mass limits. Our study comprises Tolman-Oppenheimer-Volkoff solutions of
relativistic objects with non-uniform energy densities. The results leave open
some questions concerning stable regular solutions that, to an external
observer, very closely reproduce the preliminary conditions to form
Schwarzschild black holes.
