Existence of Magnetic Compressible Fluid Stars

The existence of magnetic star solutions which are axi-symmetric stationary solutions for the Euler-Poisson system of compressible fluids coupled to a magnetic field is proved in this paper by a variational method. Our method of proof consists of deriving an elliptic equation for the magnetic potential in cylindrical coordinates in $\mathbb{R}^3$, and obtaining the estimates of the Green's function for this elliptic equation by transforming it to 5-Laplacian.