  We investigate a stationary pair production cascade in the outer
magnetosphere of a spinning neutron star. The charge depletion due to global
flows of charged particles, causes a large electric field along the magnetic
field lines. Migratory electrons and/or positrons are accelerated by this field
to radiate gamma-rays via curvature and inverse-Compton processes. Some of such
gamma-rays collide with the X-rays to materialize as pairs in the gap. The
replenished charges partially screen the electric field, which is
self-consistently solved together with the energy distribution of particles and
gamma-rays at each point along the field lines. By solving the set of Maxwell
and Boltzmann equations, we demonstrate that an external injection of charged
particles at nearly Goldreich-Julian rate does not quench the gap but shifts
its position and that the particle energy distribution cannot be described by a
power-law. The injected particles are accelerated in the gap and escape from it
with large Lorentz factors. We show that such escaping particles migrating
outside of the gap contribute significantly to the gamma-ray luminosity for
young pulsars and that the soft gamma-ray spectrum between 100 MeV and 3 GeV
observed for the Vela pulsar can be explained by this component. We also
discuss that the luminosity of the gamma-rays emitted by the escaping particles
is naturally proportional to the square root of the spin-down luminosity.
