  Equilibrium configurations for a self-gravitating scalar field with
self-interaction are constructed. The corresponding Schr\"odinger-Poisson (SP)
system is solved using finite differences assuming spherical symmetry. It is
shown that equilibrium configurations of the SP system are late-time attractor
solutions for initially quite arbitrary density profiles, which relax and
virialize through the emission of scalar field bursts; a process dubbed
gravitational cooling. Among other potential applications, these results
indicate that scalar field dark matter models (in its different flavors)
tolerate the introduction of a self-interaction term in the SP equations. This
study can be useful in exploring models in which dark matter in galaxies is not
point-like.
