Positive solutions of certain nonlinear elliptic systems with self-diffusions: nondegenerate vs. degenerate diffusions

We discuss the existence of positive solutions to certain strongly-coupled nonlinear elliptic systems with self-diffusions under homogeneous Dirichlet boundary conditions. Using the global positive coexistence results for predator-prey, competition and symbiotic interactions between two species, sufficient conditions for the positive solutions of the degenerate self-diffusive systems are studied. We also investigate the local behavior, namely, the local existence, uniqueness and stability, of positive solutions for predator-prey and competition interactions. Our method is based on the decoupling technique and bifurcation theory.