  We consider one- and two-dimensional (1D and 2D) optical or matter-wave media
with a maximum of the local self-repulsion strength at the center, and a
minimum at periphery. If the central area is broad enough, it supports ground
states in the form of flat-floor \textquotedblleft bubbles", and topological
excitations, in the form of dark solitons in 1D and vortices with winding
number $m$ in 2D. Unlike bright solitons, delocalized bubbles and dark modes
were not previously considered in this setting. The ground and excited states
are accurately approximated by the Thomas-Fermi expressions. The 1D and 2D
bubbles, as well as vortices with $m=1$, are completely stable, while the dark
solitons and vortices with $m=2$ have nontrivial stability boundaries in their
existence areas. Unstable dark solitons are expelled to the periphery, while
unstable double vortices split in rotating pairs of unitary ones. Displaced
stable vortices precess around the central point.

Pattern Formation and Solitons;Optics