  We discuss the effects of resonant $\nu_e \to \nu_s$ and $\bar{\nu}_e \to
\bar{\nu}_s$ ($\nu_s$ is a sterile neutrino) conversions in the dense medium of
a supernova. In particular, we assume the sterile neutrino $\nu_s$ to be in the
hot dark matter few eV mass range. The implications of such a scenario for the
supernova shock re-heating, the detected $\bar\nu_e$ signal from SN1987A and
for the r-process nucleosynthesis hypothesis are analysed in some detail. The
resulting constraints on mixing and mass difference for the $\nu_e-\nu_s$
system are derived. There is also an allowed region in the neutrino parameter
space for which the r-process nucleosynthesis can be enhanced.
