Highly nonlinear large-competition limits of elliptic systems

This article is concerned with the effect of different types of competitive interaction terms on the large-interaction limit of nonlinear elliptic systems modelling the steady states of populations that compete in some region. As the competition rate tends to infinity, we show that non-negative solutions of quite simple-looking systems converge to the positive and negative parts of a solution of a scalar limit problem which may be much more strongly nonlinear than the original system, possibly with quadratic growth in the gradient of the limit function.