  We obtain classification, solvability and nonexistence theorems for positive
stationary states of reaction-diffusion and Schr\"odinger systems involving a
balance between repulsive and attractive terms. This class of systems contains
PDE arising in biological models of Lotka-Volterra type, in physical models of
Bose-Einstein condensates and in models of chemical reactions. We show, with
different proofs, that the results obtained in [ARMA, 213 (2014), 129-169] for
models with homogeneous diffusion are valid for general heterogeneous media,
and even for controlled inhomogeneous diffusions.
