  We study the plasma branch of an homogeneous three-dimensional electron gas
in an $s$-wave superconducting state. We focus on the regime where the plasma
frequency $\omega_p$ is comparable to the gap $\Delta$, which is experimentally
realized in cuprates. Although a sum rule guarantees that the departure of the
plasma branch always coincides with the plasma frequency, the dispersion and
lifetime of the plasmons is strongly affected by the presence of the pair
condensate, especially at energies close to the pair-breaking threshold
$2\Delta$. When $\omega_p $ is above $1.7\Delta $, the level repulsion is
strong enough to give the plasma branch an anomalous, negative dispersion with
a minimum at finite wavelength. At non-zero temperature and at
$\omega_p>2\Delta$, we treat in a non-perturbative way the coupling of plasmons
to the fermionic excitations, and show that a broadened plasma resonance inside
the pair-breaking continuum coexists with an undamped solution in the band gap.
This resonance splitting is associated with the presence of multiple poles in
the analytic continuation of the propagator of the Cooper pairs.
