  We consider compact boson stars that arise for a V-shaped scalar field
potential. They represent a one parameter family of solutions of the scaled
Einstein-signum-Gordon equations. We analyze the physical properties of these
solutions and determine their domain of existence. Along their physically
relevant branch emerging from the compact Q-ball solution, their mass increases
with increasing radius. Empoying arguments from catastrophe theory we argue
that this branch is stable, until the maximal value of the mass is reached.
There the mass and size are on the order of magnitude of the Schwarzschild
limit, and thus the spiralling respectively oscillating behaviour, well-known
for compact stars, sets in.
