Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation

TL;DR

This paper proves that positive solutions with algebraic growth to a specific elliptic system exhibit monotonicity and one-dimensional symmetry in all dimensions, confirming a conjecture related to the system's symmetry properties.

Contribution

It establishes monotonicity and symmetry results for solutions of an elliptic system arising in Bose-Einstein condensation, confirming a conjecture by Berestycki et al.

Findings

Solutions are monotone in certain directions.

Solutions are one-dimensional symmetric.

The Gibbons-type conjecture is proven.

Abstract

We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: \[ \begin{cases} -\Delta u = -u v^2 & \text{in $\R^N$}\\ -\Delta v= -u^2 v & \text{in $\R^N$}, \end{cases} \] for every dimension $N \ge 2$. In particular, we prove a Gibbons-type conjecture proposed by H. Berestycki, T. C. Lin, J. Wei and C. Zhao.