  We consider boson star solutions in a $D$-dimensional, asymptotically anti-de
Sitter spacetime and investigate the influence of the cosmological term on
their properties. We find that for $D>4$ the boson star properties are close to
those in four dimensions with a vanishing cosmological constant. A different
behavior is noticed for the solutions in the three dimensional case. We
establish also the non-existence of static, spherically symmetric black holes
with a harmonically time-dependent complex scalar field in any dimension
greater than two.
