  We investigate systematically the effect of the nonlinear correction to the
usual Maxwell electrodynamics on the holographic dual models in the backgrounds
of AdS black hole and AdS soliton. Considering three types of typical nonlinear
electrodynamics, we observe that in the black hole background the higher
nonlinear electrodynamics correction makes the condensation harder to form and
changes the expected relation in the gap frequency, which is similar to that
caused by the curvature correction. However, in strong contrast to the
influence of the curvature correction, we find that in the AdS soliton
background the nonlinear electrodynamics correction will not affect the
properties of the holographic superconductor and insulator phase transitions,
which may be a quite general feature for the s-wave holographic
superconductor/insulator system.
