  In this paper, the existence and uniqueness of strong axisymmetric solutions
with large flux for the steady Navier-Stokes system in a pipe are established
even when the external force is also suitably large in $L^2$. Furthermore, the
exponential convergence rate at far fields for the arbitrary steady solutions
with finite $H^2$ distance to the Hagen-Poiseuille flows is established as long
as the external forces converge exponentially at far fields. The key point to
get the existence of these large solutions is the refined estimate for the
derivatives in the axial direction of the stream function and the swirl
velocity, which exploits the good effect of the convection term. An important
observation for the asymptotic behavior of general solutions is that the
solutions are actually small at far fields when they have finite $H^2$ distance
to the Hagen-Poiseuille flows. This makes the estimate for the linearized
problem play a crucial role in studying the convergence of general solutions at
far fields.
