On Phase-Separation Models: Asymptotics and Qualitative Properties

In this paper we study bound state solutions of a class of two-component nonlinear elliptic systems with a large parameter tending to infinity. The large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface. To obtain the profile of bound state solutions near the interface, we prove the uniform Lipschitz continuity of bound state solutions when the spatial dimension is N = 1. Furthermore, we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity. These unbounded solutions are useful for rigorously deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of Du and Zhang (Discontin Dynam Sys, 2012). When the spatial dimension is N = 2, we establish the De Giorgi type conjecture for the blow-up nonlinear elliptic system under suitable conditions at infinity on bound state solutions. These results naturally lead us to formulate De Giorgi type conjectures for these types of systems in higher dimensions.