On Entire Solutions of an Elliptic System Modeling Phase Separations

TL;DR

This paper investigates the qualitative properties of solutions to a limiting elliptic system modeling phase separation in Bose-Einstein condensates, including uniqueness, stability, and construction of solutions with polynomial growth.

Contribution

It establishes uniqueness in one dimension, characterizes stable solutions in two dimensions, and constructs multi-component solutions with polynomial growth, extending previous results.

Findings

Unique one-dimensional profile in 1D

Stable solutions with linear growth are one-dimensional in 2D

Constructed entire solutions with polynomial growth in 2D and extended to multi-component systems

Abstract

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