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

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.