hairy ads black holes with robin boundary conditions

We study hairy black holes in einstein-maxwell-complex scalar theory in four-dimensional asymptotically global anti-de sitter (ads) spacetime when the robin boundary conditions are imposed on the scalar field. the hairy solutions branch from the reissner-nordstr&{o}m-ads (rnads) black holes at the onset of instability of the scalar field under the robin boundary conditions. there are also associated horizonless solutions called boson stars. comparing thermal ads, rnads, charged boson stars, and hairy black holes, we obtain phase diagrams in the grand canonical ensemble.