  We study the phenomenon of cavitation for the displacement boundary value
problem of radial, isotropic compressible elasticity for a class of stored
energy functions of the form $W(F) + h(\det F)$, where $W$ grows like
$||F||^n$, and $n$ is the space dimension. In this case it follows (from a
result of Vodopyanov, Goldshtein and Reshetnyak) that discontinuous
deformations must have infinite energy. After characterizing the rate at which
this energy blows up, we introduce a modified energy functional which differs
from the original by a null lagrangian, and for which cavitating energy
minimizers with finite energy exist. In particular, the Euler--Lagrange
equations for the modified energy functional are identical to those for the
original problem except for the boundary condition at the inner cavity. This
new boundary condition states that a certain modified radial Cauchy stress
function has to vanish at the inner cavity. This condition corresponds to the
radial Cauchy stress for the original functional diverging to $-\infty$ at the
cavity surface. Many previously known variational results for finite energy
cavitating solutions now follow for the modified functional, such as the
existence of radial energy minimizers, satisfaction of the Euler-Lagrange
equations for such minimizers, and the existence of a critical boundary
displacement for cavitation. We also discuss a numerical scheme for computing
these singular cavitating solutions using regular solutions for punctured
balls. We show the convergence of this numerical scheme and give some numerical
examples including one for the incompressible limit case. Our approach is
motivated in part by the use of the renormalized energy for Ginzberg-Landau
vortices.
