  The structure of steady axisymmetric force-free magnetosphere of a Kerr black
hole (BH) is governed by a second-order partial differential equation of
$A_\phi$ depending on two "free" functions $\Omega(A_\phi)$ and $I(A_\phi)$,
where $A_\phi$ is the $\phi$ component of the vector potential of the
electromagnetic field, $\Omega$ is the angular velocity of the magnetic field
lines and $I$ is the poloidal electric current. In this paper, we investigate
the solution uniqueness. Taking asymptotically uniform field as an example,
analytic studies imply that there are infinitely many solutions approaching
uniform field at infinity, while only a unique one is found in general
relativistic magnetohydrodynamic simulations. To settle down the disagreement,
we reinvestigate the structure of the governing equation and numerically solve
it with given constraint condition and boundary condition. We find that the
constraint condition (field lines smoothly crossing the light surface (LS)) and
boundary conditions at horizon and at infinity are connected via radiation
conditions at horizon and at infinity, rather than being independent. With
appropriate constraint condition and boundary condition, we numerically solve
the governing equation and find a unique solution. Contrary to naive
expectation, our numerical solution yields a discontinuity in the angular
velocity of the field lines and a current sheet along the last field line
crossing the event horizon. We also briefly discuss the applicability of the
perturbation approach to solving the governing equation.
