  We introduce a one-dimensional (1D) spatially inhomogeneous Bose-Hubbard
model (BHM) with the strength of the onsite repulsive interactions growing,
with the discrete coordinate $z_{j}$, as $|z_{j}|^{\alpha }$ with $\alpha >0$.
Recently, the analysis of the mean-field (MF) counterpart of this system has
demonstrated self-trapping of robust unstaggered discrete solitons, under
condition $\alpha >1$. Using the numerically implemented method of the density
matrix renormalization group (DMRG), we demonstrate that, in a certain range of
interaction, the BHM also self-traps, in the ground state, into a soliton-like
configuration, at $\alpha >1$, and remains weakly localized at $\alpha <1$. An
essential quantum feature is a residual density in the background surrounding
the soliton-like peak in the BHM ground state, while in the MF limit the
finite-density background is absent. Very strong onsite repulsion eventually
destroys soliton-like states, and, for integer densities, the system enters the
Mott phase with a spatially uniform density
