Semiclassical asymptotic behavior of ground state for the two‐component Hartree system

We consider the semiclassical asymptotic behaviors of ground state solution for the following two‐component Hartree system: which is originated from the study on cold atoms of boson and fermion system with long‐range interaction. Under the assumption by detailed compactness analysis, we prove that there is a β₀>0 such that if β<β₀, the system has a ground state solution. For this solution, the energy estimates and the decay rates are presented, and the asymptotic profiles as ε→0 are displayed in details for β<0 and β>0, respectively. Furthermore, we show that for β<0, the phase separation phenomenon may occur.