A trapping principle for discontinuous elliptic systems of mixed monotone type

We consider discontinuous semilinear elliptic systems, with boundary conditions on the individual components of Dirichlet/Neumann type. The system is a divergence form generalization of Δu = f(·,u). The components of f are required to satisfy monotonicity conditions associated with competitive or cooperative species. The latter model defines a system of mixed monotone type. We also illustrate the theory via higher order mixed monotone systems which combine competitive and cooperative subunits. We seek solutions on special intervals defined by lower and upper solutions associated with outward pointing vector fields. It had been shown by Heikkilä and Lakshmikantham that the general discontinuous mixed monotone system does not necessarily admit solutions on an interval defined by lower and upper solutions. Our result, obtained via the Tarski fixed-point theorem, shows that solutions exist for the models described above in the sense of a measurable selection (in the principal arguments) from a maximal monotone multivalued mapping. We use intermediate variational inequalities in the proof. Applications involving quantum confinement and chemically reacting systems with change of phase are discussed. These are natural examples of discontinuous systems.