  Searches for dark matter imprints are one of the most active areas of current
research. We focus here on light fields with mass $m_B$, such as axions and
axion-like candidates. Using perturbative techniques and full-blown nonlinear
Numerical Relativity methods, we show that (i) dark matter can pile up in the
center of stars, leading to configurations and geometries oscillating with
frequency which is a multiple of f=$2.5 10^{14}$ $m_B c^2$/eV Hz. These
configurations are stable throughout most of the parameter space, and arise out
of credible mechanisms for dark-matter capture. Stars with bosonic cores may
also develop in other theories with effective mass couplings, such as
(massless) scalar-tensor theories. We also show that (ii) collapse of the host
star to a black hole is avoided by efficient gravitational cooling mechanisms.
