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 axionlike candidates. Using perturbative techniques and full-blown nonlinear numerical relativity methods, we show the following. (i) Dark matter can pile up in the center of stars, leading to configurations and geometries oscillating with a frequency that is a multiple of f = 2.5 x 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.