  We study, analytically and numerically, the stationary states in the system
of two linearly coupled nonlinear Schr{\"o}dinger equations in two spatial
dimensions, with the nonlinear interaction coefficients of opposite signs. This
system is the two-dimensional analog of the coupled-mode equations for a
condensate in the double-well trap [\textit{Physical Review A} \textbf{69},
033609 (2004)]. In contrast to the one-dimensional case, where the bifurcation
from zero leads to stable bright solitons, in two spatial dimensions this
bifurcation results in the appearance of unstable soliton solutions (the
Townes-type solitons). With the use of a parabolic potential the ground state
of the system is stabilized. It corresponds to strongly coupled condensates and
is stable with respect to collapse. This is in sharp contrast to the
one-dimensional case, where the ground state corresponds to weakly coupled
condensates and is unstable. Moreover, the total number of atoms of the stable
ground state can be much higher than the collapse threshold for a single
two-dimensional condensate with a negative s-wave scattering length.
