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

Abstract

We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: \[ \begin{cases} -Δu = -u v^2 & \text{in ${\mathbb{R}}^N$}\\ -Δv= -u^2 v & \text{in ${\mathbb{R}}^N$}, \end{cases} \] 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.

References in corpus (2)

• The dynamics of component separation in a binary mixture of Bose-Einstein condensates
• Entire solutions with exponential growth for an elliptic system modeling phase-separation

Cited by in corpus (5)

• Uniform bounds for strongly competing systems: the optimal Lipschitz case
• Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation
• On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects
• Multidimensional entire solutions for an elliptic system modelling phase separation
• On the Asymptotic Growth of Positive Solutions to a Nonlocal Elliptic Blow-up System Involving Strong Competition