  In this paper we look for minimizers of the energy functional for isotropic
compressible elasticity taking into consideration the effect of a gravitational
field induced by the body itself. We consider the displacement problem in which
the outer boundary of the body is subjected to a Dirichlet type boundary
condition. For a spherically symmetric body occupying the unit ball
$\mathcal{B}\in\mathbb{R}^3$, the minimization is done within the class of
radially symmetric deformations. We give conditions for the existence of such
minimizers, for satisfaction of the Euler--Lagrange equations, and show that
for large displacements the minimizer must develop a cavity at the centre. A
numerical scheme for approximating these minimizers is given together with some
simulations that show the dependence of the cavity radius and minimum energy on
the displacement and mass density of the body.
