  In this work, expanded solutions of force-free magnetospheres on general Kerr
black holes are derived through a radial distance expansion method. From the
regular conditions both at the horizon and at spatial infinity, two previously
known asymptotical solutions (one of them is actually an exact solution) are
identified as the only solutions that satisfy the same conditions at the two
boundaries. Taking them as initial conditions at the boundaries, expanded
solutions up to the first few orders are derived by solving the stream equation
order by order. It is shown that our extension of the exact solution can
(partially) cure the problems of the solution: it leads to magnetic domination
and a mostly timelike current for restricted parameters.
