  Emerets's experiments on pressurized sulfur hydride have shown that H3S metal
has the highest known superconducting critical temperature Tc=203K. The Emerets
data show pressure induced changes of the isotope coefficient between 0.25 and
0.5, in disagreement with Eliashberg theory which predicts a nearly constant
isotope coefficient. We assign the pressure dependent isotope coefficient to
Lifshitz transitions induced by pressure and zero point lattice fluctuations.
It is known that pressure could induce changes of the topology of the Fermi
surface, called Lifshitz transitions, but were neglected in previous papers on
the H$_3$S superconductivity issue. Here we propose that H3S is a multi-gap
superconductor with a first condensate in the BCS regime (in the large Fermi
surface with high Fermi energy) which coexists with a second condensates in the
BCS-BEC crossover regime (located on a small Fermi surface spots with small
Fermi energy) near the $\Gamma$ and M point. We discuss the need of
Bianconi-Perali-Valletta (BPV) superconductivity theory for superconductivity
in H3S. It includes both the correction of the chemical potential due to
pairing and the configuration interaction between different condensates,
neglected by the Eliashberg theory. Here the shape resonance in superconducting
gaps, similar to Feshbach resonance in ultracold gases, gives a relevant
contribution to amplify the critical temperature. Therefore this work provides
some key tools needed in the search for new room temperature superconductors.
