On Entire Solutions of an Elliptic System Modeling Phase Separations

Details On Entire Solutions of an Elliptic System Modeling Phase Separations On Entire Solutions of an Elliptic System Modeling Phase Separations

2012-04-04

arxiv.org/abs/1204.1038v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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