  We study the force-free electrodynamics on rotating black holes in the
Born-Infeld (BI) effective theory. The stream equation describing a steady and
axisymmetric magnetosphere is derived. From its near-horizon behavior, we
obtain the modified Znajek regularity condition, with which we find that the
horizon resistivity in the BI theory is generally not a constant. As expected,
the outer boundary condition far away from the hole remains unchanged. In terms
of the conditions at both boundaries, we derive the perturbative solution of
split monopole in the slow rotation limit. It is interesting to realise that
the correction to the solution relies not only on the parameter in the BI
theory, but also on the radius (or the mass) of the hole. We also show that the
quantum effects can undermine the energy extraction process of the
magnetosphere in the non-linear theory and the extraction rate gets the maximum
in the Maxwell theory.
