We investigate the positive coexistence to certain strongly-coupled nonlinear
elliptic systems with self-cross diffusions under homogeneous Robin boundary
conditions. Competing interactions between two species are considered. Conditions
of the positive coexistence to self-cross diffusive systems can be expressed in terms of
the spectral property of differential operators of nonlinear Schrödinger type which reflect
the influence of the domain and nonlinearity in the system. Decoupling method
and nonlinear fixed point theorem are employed.