Minimization problems in L1(R3)

In this paper, minimization problems in L¹($\text{R}$³) are considered. These problems arise in astrophysics for the determination of equilibrium configurations of axially symmetric rotating fluids (rotating stars). Under nearly optimal assumptions a minimizer is proved to exist by a direct variational method, which heavily uses the symmetry of the problem in order to get some compactness. Finally, by looking directly at the Euler equation, we give some existence results (of solutions of the Euler equation) even if the infimum is not finite.