  Huge electromagnetic fields are known to be present during the late stages of
the dynamics of supernovae. Thus, when dealing with electrodynamics in this
context, the possibility may arise to probe nonlinear theories (generalizations
of the Maxwellian electromagnetism). We firstly solve Einstein field equations
minimally coupled to an arbitrary (current-free) nonlinear Lagrangian of
electrodynamics (NLED) in the slow rotation regime $a\ll M$ (black hole's
mass), up to first order in $a/M$. We then make use of the robust and
self-contained Born-Infeld Lagrangian in order to compare and contrast the
physical properties of such NLED spacetime with its Maxwellian counterpart (a
slowly rotating Kerr-Newman spacetime), especially focusing on the astrophysics
of both neutrino flavor oscillations ($\nu_e \rightarrow \nu_\mu, \nu_\tau$)
and spin-flip ($\nu_l \rightarrow \nu_r$, "$l$" stands for "left" and "$r$"
stands for "right", change of neutrino handedness) mass level-crossings, the
equivalent to gyroscopic precessions. Such analysis proves that in the
spacetime of a slowly rotating nonlinear charged black hole (RNCBH),
intrinsically associated with the assumption the electromagnetism is nonlinear,
the neutrino dynamics in core-collapse supernovae could be significantly
changed. In such astrophysical environment a positive enhancement (reduction of
the electron fraction $Y_e<0.5$) of the r-process may take place. Consequently,
it might result in hyperluminous supernova explosions due to enlargement, in
atomic number and amount, of the decaying nuclides. Finally, we envisage some
physical scenarios that may lead to short-lived charged black holes with high
charge-to-mass ratios (associated with unstable highly magnetized neutron
stars) and ways to possibly disentangle theories of the electromagnetism from
other black holes observables (by means of light polarization measurements).
