  Quantum electrodynamics (QED) effects may be included in physical processes
of magnetar and pulsar magnetospheres with strong magnetic fields. Involving
the quantum corrections, the Maxwell electrodynamics is modified to non-linear
electrodynamics. In this work, we study the force-free magnetosphere in
non-linear electrodynamics in a general framework. The pulsar equation
describing a steady and axisymmetric magnetosphere is derived, which now admits
solutions with corrections. We derive the first-order non-linear corrections to
the near-zone dipole magnetosphere in some popular non-linear effective
theories. The field lines of the corrected dipole tend to converge on the
rotational axis so that the fields in the polar region are stronger compared to
the pure dipole case.
