  We study a system of penetrable bosons embedded in a spherical surface. Under
the assumption of weak interaction between the particles, the ground state of
the system is, to a good approximation, a pure condensate. We employ
thermodynamic arguments to investigate, within a variational ansatz for the
single-particle state, the crossover between distinct finite-size "phases" in
the parameter space spanned by the sphere radius and the chemical potential. In
particular, for radii up to a few interaction ranges we examine the stability
of the fluid phase with respect to a number of crystal-like arrangements having
the symmetry of a regular or semi-regular polyhedron. We find that, while
quantum fluctuations keep the system fluid at low density, upon compression it
eventually becomes inhomogeneous, i.e., particles gather together in clusters.
As the radius increases, the nature of the high-density aggregate varies and we
observe a sequence of transitions between different cluster phases ("solids"),
whose underlying rationale is to maximize the coordination number of clusters,
while ensuring at the same time the proper distance between each neighboring
pair. We argue that, at least within our mean-field description, every cluster
phase is supersolid.
