On Entire Solutions of an Elliptic System Modeling Phase Separations

Abstract

We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: Δu=u v^2 in R^n, Δv= v u^2 in R^n, u, v>0\quad in R^n. 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 ${\mathbb{R}}^2$ with polynomial growth $|x{|}^d$ for any positive integer $d\ge 1$. For $d\ge 2$, these solutions are not one-dimensional. The construction is also extended to multi-component elliptic systems.