Spherically symmetric equilibria for self-gravitating kinetic or fluid
  models in the non-relativistic and relativistic case - A simple proof for
  finite extension
  We consider a self-gravitating collisionless gas as described by the
Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as
described by the Euler-Poisson or Einstein-Euler system. We give a simple proof
for the finite extension of spherically symmetric equilibria, which covers all
these models simultaneously. In the Vlasov case the equilibria are
characterized by a local growth condition on the microscopic equation of state,
i.e., on the dependence of the particle distribution on the particle energy, at
the cut-off energy E_0, and in the Euler case by the corresponding growth
condition on the equation of state p=P(\rho) at \rho=0. These purely local
conditions are slight generalizations to known such conditions.
