  We consider a two-dimensional (2D) generalization of a recently proposed
model [Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete
solitons supported by the defocusing nonlinearity whose local strength grows
from the center to the periphery. We explore the 2D model starting from the
anti-continuum (AC) limit of vanishing coupling. In this limit, we can
construct a wide variety of solutions including not only single-site
excitations, but also dipole and quadrupole ones. Additionally, two separate
families of solutions are explored: the usual "extended" unstaggered bright
solitons, in which all sites are excited in the AC limit, with the same sign
across the lattice (they represent the most robust states supported by the
lattice, their 1D counterparts being what was considered as 1D bright solitons
in the above-mentioned work), and the vortex cross, which is specific to the 2D
setting. For all the existing states, we explore their stability (analytically,
whenever possible). Typical scenarios of instability development are exhibited
through direct simulations.
