  Self-interacting boson stars have been shown to alleviate the astrophysically
low maximal mass of their non-self-interacting counterparts. We report some
physical features of spinning self-interacting boson stars, namely their
compactness, the occurence of ergo-regions and the scalar field profiles, for a
sample of values of the coupling parameter. The results agree with the general
picture that these boson stars are comparatively less compact than the
non-self-interacting ones. We also briefly discuss the effect of scalar
self-interactions on the properties of Kerr black holes with scalar hair.
