On entire solutions of an elliptic system modeling phase separations

We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: {Δu=uv2in Rn,Δv=vu2in Rn,u,v>0in Rn. When n=1, we prove uniqueness of the one-dimensional profile. In dimension 2, we prove that stable solutions with linear growth must be one-dimensional. Then we construct entire solutions in R2 with polynomial growth ∣x∣d for any positive integer d≥1. For d≥2, these solutions are not one-dimensional. The construction is also extended to multi-component elliptic systems.