On Rotating Axisymmetric Solutions of the Euler-Poisson Equations

We consider stationary axisymmetric solutions of the Euler-Poisson equations, which govern the internal structure of barotropic gaseous stars. We take the general form of the equation of states which cover polytropic gaseous stars indexed by $6/5<\gamma <2$ and also white dwarfs. A generic condition of the existence of stationary solutions with differential rotation is given, and the existence of slowly rotating configurations near spherically symmetric equilibria is shown. The problem is formulated as a nonlinear integral equation, and is solved by an application of the infinite dimensional implicit function theorem. Oblateness of star surface is shown and also relationship between the central density and the total mass is given.