  We have performed density functional calculations in conjunction with the
linearized Migdal-Eliashberg equations and the functional derivative approach,
which takes into account the energy-range dependence of the density of states
at the Fermi level ($N(\varepsilon)$ variable on the scale of phonon energy),
to determine the evolution of the critical temperature ($T_{c}$) and the
isotope effect coefficient ($\alpha$) of H$_{3}$S as a function of pressure on
its $lm\bar{3}m$ crystal-structure range (from 162 to 250 GPa). Such approach,
in comparison with $N(\varepsilon)$=$N(0)$=const., improves the agreement of
$T_{c}$ with available experiments on the whole range of studied pressure.
Considering for $\alpha$ two main contributions: one positive coming from the
electron-phonon (el-ph) and the other negative from the electron-electron
interaction (el-el), we obtained a monotonic decrement as a function of
pressure, independent of applied scheme ($N(\varepsilon)$ or $N(0)$). However,
when $N(\varepsilon)$ is taken into account, an important renormalization
occurs on both contributions, el-ph and el-el, improving the agreement with
experimental data, specially for the high-pressure regime. The observed
evolution of $T_{c}$ and $\alpha$ as a function of pressure indicates, thus,
the crucial role of the energy-dependence on $N(\varepsilon)$ for the proper
analysis and description of the superconducting state on high-$T_{c}$ metal
hydrides as H$_{3}$S, by considering the role of its van Hove singularities.
