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arXiv · 1204.1038

On Entire Solutions of an Elliptic System Modeling Phase Separations

Henri Berestycki·Susanna Terracini·Kelei Wang·Juncheng Wei
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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.

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Henri Berestycki, Susanna Terracini, Kelei Wang, Juncheng Wei. 2012-04-04. On Entire Solutions of an Elliptic System Modeling Phase Separations. https://arxiv.org/abs/1204.1038

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