  We consider scalar dark matter models where the theory has a shift symmetry
only broken by the scalar mass term. We restrict ourselves to K-essence kinetic
terms where the shift symmetric part of the Lagrangian is a function of the
first derivatives of the scalar field only. In the low-amplitude and
nonrelativistic regime, which applies on large galactic scales, scalar clouds
form solitons with a finite core. Close to the center of galaxies, where a
supermassive Black Hole (BH) resides, we analyze the scalar field distribution
and the fate of the dark matter soliton when subject to the BH gravitational
attraction. We show that the scalar field profile around such a central BH can
be described by new oscillatory solutions of a modified Klein-Gordon equation,
which generalize the harmonic oscillations of free scalar dark matter in a flat
environment and the Jacobi elliptic functions of the $\phi^4$ model. Moreover,
we find that, depending on the form of the K-essence kinetic term, regular
solutions can be constructed or not, which connect the relativistic ingoing
wavelike profile of the scalar field at the BH horizon to the nearly static
nonrelativistic soliton at large distance. These profiles have a constant flux
and represent the slow infall of scalar matter into the BH. We show that this
regular behavior is only possible for K-essence functions that satisfy the
usual conditions for the absence of ghosts and gradient instabilities, together
with a new restriction on the growth of the kinetic function $K(X)$ for large
argument. It turns out that the same conditions of stability guarantee that
quantum corrections are tamed, provided that the mass of the scalar field is
less than $10^{-3}$ eV and the strong coupling scale of the model $\Lambda$ is
much larger than the scalar mass.
