  Context: It has been suggested that some supernovae (SNe) may be powered by a
magnetar formed at the moment of the explosion. While this scenario has mostly
been applied to hydrogen-free events, it may be possible also for hydrogen-rich
objects. Aims: We explore the effect of including a magnetar on the light
curves of supernovae with H-rich progenitors. Methods: We have applied a
version of our one-dimensional LTE radiation hydrodynamics code that takes into
account the relativistic motion of the ejecta caused by the extra energy
provided by the magnetar. For a fixed red supergiant (RSG) progenitor, we have
obtained a set of light curves that corresponds to different values of the
magnetar initial rotation energy and the spin-down timescale. The model is
applied to SN~2004em and OGLE-2014-SN-073, two peculiar Type II SNe with
long-rising SN1987A-like light curves, although with much larger luminosities.
Results: The presence of a plateau phase in either normal or superluminous
supernovae is one possible outcome, even if a magnetar is continuously
injecting energy into the ejecta.In other cases, the light curve shows a peak
but not a plateau. Also, there are intermediate events with a first peak
followed by a slow decline and a late break of the declining slope. Our models
show that bright and long rising morphologies are possible even assuming RSG
structures. Conclusions: A large number of supernova discoveries per year
reveal unexpected new types of explosions. According to our results, SLSNe II-P
are to be expected, as well as a variety of light curve morphologies that can
all be possible signs of a newly born magnetar.
