Existence of axially symmetric weak solutions to steady MHD with non-homogeneous boundary conditions

We establish the existence of axially symmetric weak solutions to steady incompressible magnetohydrodynamics with non-homogeneous boundary conditions. The key issue is the Bernoulli's law for the total head pressure $\Phi=\f 12(|{\bf u}|^2+|{\bf h}|^2)+p$ to a special class of solutions to the inviscid, non-resistive MHD system, where the magnetic field only contains the swirl component.