  We demonstrate that modulation of the local strength of the cubic
self-focusing (SF) nonlinearity in the two-dimensional (2D) geometry, in the
form of a circle with contrast $\Delta g$ of the SF coefficient relative to the
ambient medium with a weaker nonlinearity, stabilizes a family of fundamental
solitons against the critical collapse. The result is obtained in an analytical
form, using the variational approximation (VA) and Vakhitov-Kolokolov (VK)
stability criterion, and corroborated by numerical computations. For the small
contrast, the stability interval of the soliton's norm scales as $\Delta N\sim
\Delta g$ (the replacement of the circle by an annulus leads to a reduction of
the stability region by perturbations breaking the axial symmetry). To further
illustrate this mechanism, we demonstrate, in an exact form, the stabilization
of 1D solitons against the critical collapse under the action of a locally
enhanced quintic SF nonlinearity.
