  We consider boson stars and black holes in scalar electrodynamics with a
V-shaped scalar potential. The boson stars come in two types, having either
ball-like or shell-like charge density. We analyze the properties of these
solutions and determine their domains of existence. When mass and charge become
equal, the space-times develop a throat. The shell-like solutions need not be
globally regular, but may possess a horizon. The space-times then consist of a
Schwarzschild-type black hole in the interior, surrounded by a shell of charged
matter, and thus a Reissner-Nordstroem-type space-time in the exterior. These
solutions violate black hole uniqueness. The mass of the black hole solutions
is related to the mass of the regular shell-like solutions by a mass formula of
the type first obtained within the isolated horizon framework.
