Uniqueness of constraint minimizers for coupled planar Gross-Pitaevskii systems with logarithmic convolutions

Yujin Guo; Min Liu; Shu Zhang

doi:10.3934/dcds.2025136

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2026, Volume 48: 64-103. Doi: 10.3934/dcds.2025136

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Uniqueness of constraint minimizers for coupled planar Gross-Pitaevskii systems with logarithmic convolutions

1.
School of Mathematics and Statistics, Key Laboratory of Nonlinear Analysis & Applications (Ministry of Education), Central China Normal University, P.O. Box 71010, Wuhan 430079, China
2.
School of Mathematics, Liaoning Normal University, Dalian 116029, China
3.
School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

^*Corresponding author: Min Liu
^*Corresponding author: Min Liu

Received: May 2025

Revised: August 2025

Early access: August 25, 2025

Published online: August 25, 2025

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Abstract

This paper is concerned with the $ L^2 $-critical minimization problem of the coupled planar Gross-Pitaevskii system with a harmonic potential and logarithmic convolutions. We first establish the existence and nonexistence of constraint minimizers for the system, which is connected with the unique positive solution of $ -\Delta u+u-u^3 = 0 $ in $ \mathbb{R}^2 $. The refined limiting behavior and exponential decay of nonnegative constraint minimizers are also analyzed. We further obtain the uniqueness of nonnegative constraint minimizers by overcoming the non-invariance under translations of the potential and the sign-changing property of the logarithmic convolutions.

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Mathematics Subject Classification: Primary: 35A02, 35J50; Secondary: 35Q40.

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