  In this paper we study the linear stability of selfinteracting boson stars in
the nonrelativistic limit of the Einstein-Klein-Gordon theory. For this
purpose, based on a combination of analytic and numerical methods, we determine
the behavior of general linear perturbations around the stationary and
spherically symmetric solutions of the Gross-Pitaevskii-Poisson system. In
particular, we conclude that ground state configurations are linearly stable if
the selfinteraction is repulsive, whereas there exist a state of maximum mass
that divides the stable and the unstable branches in case the selfinteraction
is attractive. Regarding the excited states, they are in general unstable under
generic perturbations, although we identify a stability band in the first
excited states of the repulsive theory. This result is independent of the mass
of the scalar field and the details of the selfinteraction potential, and it is
in contrast to the situation of vanishing selfinteraction, in which excited
states are always unstable.
