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

siam journal on mathematical analysis

Spherically symmetric equilibria for self-gravitating kinetic or fluid  models in the non-relativistic and relativistic case - A simple proof for  finite extension