  We elaborate one- and two-dimensional (1D and 2D) models of media with
self-repulsive cubic nonlinearity, whose local strength is subject to spatial
modulation that admits the existence of flat-top solitons of various types,
including fundamental ones, 1D multipoles, and 2D vortices. Previously,
solitons of this type were only produced by models with competing
nonlinearities. The present setting may be implemented in optics and
Bose-Einstein condensates. The 1D version gives rise to an exact analytical
solution for stable flat-top solitons, and generic families may be predicted by
means of the Thomas-Fermi approximation. Stability of the obtained flat-top
solitons is analyzed by means of linear-stability analysis and direct
simulations. Fundamental solitons and 1D multipoles with $k=1$ and $2$ nodes,
as well as vortices with winding number $m=1$, are completely stable. For
multipoles with $k\geq 3$ and vortices with $m\geq 2$, alternating stripes of
stability and instability are identified in their parameter spaces.
