  The ability to gate-induce superconductivity by electrostatic charge
accumulation is a recent breakthrough in physics and nano-electronics. With the
exception of LaAlO$_3$/SrTiO$_3$ interfaces, experiments on gate-induced
superconductors have been largely confined to resistance measurements, which
provide very limited information about the superconducting state. Here, we
explore gate-induced superconductivity in MoS$_2$ by performing tunneling
spectroscopy to determine the energy-dependent density of states (DOS) for
different levels of electron density $\textit{n}$. In the superconducting
state, the DOS is strongly suppressed at energy smaller than the gap, \Delta,
which is maximum (\Delta ~ 2 meV) for $\textit{n}$ of ~ 10$^{14}$ cm$^{-2}$ and
decreases monotonously for larger $\textit{n}$. A perpendicular magnetic field
$\textit{B}$ generates states at $E<\Delta$ that fill the gap, but a 20% DOS
suppression of superconducting origin unexpectedly persists much above the
transport critical field. Conversely, an in-plane field up to 10 T leaves the
DOS entirely unchanged. Our measurements exclude that the superconducting state
in MoS$_2$ is fully gapped and reveal the presence of a DOS that vanishes
linearly with energy, the explanation of which requires going beyond a
conventional, purely phonon-driven Bardeen-Cooper-Schrieffer mechanism.
