Pinned modes in two-dimensional lossy lattices with local gain and nonlinearity.

We introduce a system with one or two amplified nonlinear sites ('hot spots', HSs) embedded into a two-dimensional linear lossy lattice. The system describes an array of evanescently coupled optical or plasmonic waveguides, with gain applied to selected HS cores. The subject of the analysis is discrete solitons pinned to the HSs. The shape of the localized modes is found in quasi-analytical and numerical forms, using a truncated lattice for the analytical consideration. Stability eigenvalues are computed numerically, and the results are supplemented by direct numerical simulations. In the case of self-focusing nonlinearity, the modes pinned to a single HS are stable and unstable when the nonlinearity includes the cubic loss and gain, respectively. If the nonlinearity is self-defocusing, the unsaturated cubic gain acting at the HS supports stable modes in a small parametric area, whereas weak cubic loss gives rise to a bistability of the discrete solitons. Symmetric and antisymmetric modes pinned to a symmetric set of two HSs are also considered.