  In the present paper we analyze and discuss some mathematical aspects of the
fluid-static configurations of a self-gravitating perfect gas enclosed in a
spherical solid shell. The mathematical model we consider is based on the
well-known Lane-Emden equation, albeit under boundary conditions that differ
from those usually assumed in the astrophysical literature. The existence of
multiple solutions requires particular attention in devising appropriate
numerical schemes apt to deal with and catch the solution multiplicity as
efficiently and accurately as possible. In sequence, we describe some
analytical properties of the model, the two algorithms used to obtain numerical
solutions, and the numerical results for two selected cases.
