  The gravitational interaction is generally considered to be too weak to be
easily submitted to systematic experimental investigation in the quantum,
microscopic, domain. In this paper we attempt to remedy this situation by
considering the gravitational influence exerted by a crystalline nanosphere of
mesoscopic size on itself, in the semi-classical, mean field, regime. We study
in depth the self-localisation process induced by the corresponding non-linear
potential of (gravitational) self-interaction. In particular, we characterize
the stability of the associated self-collapsed ground state and estimate the
magnitude of the corrections that are due to the internal structure of the
object (this includes size-effects and corrections due to the discrete, atomic,
structure of the sphere). Finally, we derive an approximated, gaussian,
dynamics which mimics several essential features of the self-gravitating
dynamics and, based on numerical results derived from this model, we propose a
concrete experimental setting which we believe might, in the foreseeable
future, reveal the existence of gravitational self-interaction effects.
