Asymptotic behaviour of solutions of planar elliptic systems with strong competition

We study a class of planar nonlinear elliptic systems with competition which includes the Hartree–Fock type approximation for a system of Bose–Einstein condensates in multiple hyperfine states as derived by Esry et al (1997 Phys. Rev. Lett. 78 3594–7). We study the limit behaviour of solutions in the case where the repulsive interaction tends to infinity and phase separation is expected. In particular, we prove the continuity of the limit shape and derive limit equations satisfied within its nodal sets. By this we complement recent work of Chang et al (2004 Physique D 196 341–61) where additional assumptions had to be made.