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:ε 2 Δ u 1 − V 1 ( x ) u 1 + μ 1 ( 1 | x | ∗ | u 1 | 2 ) u 1 + β ( 1 | x | ∗ | u 2 | 2 ) u 1 = 0 ,ε 2 Δ u 2 − V 2 ( x ) u 2 + μ 2 ( 1 | x | ∗ | u 2 | 2 ) u 2 + β ( 1 | x | ∗ | u 1 | 2 ) u 2 = 0 ,u j > 0 inR 3 ,u j → 0 as | x | → + ∞ , j = 1 , 2 ,which is originated from the study on cold atoms of boson and fermion system with long‐range interaction. Under the assumption0 < λ j : = inf R 3V j ( x ) < sup R 3V j ( x ) : = α j = lim | x | → + ∞V j ( x ) ⩽ + ∞ , j = 1 , 2 , by detailed compactness analysis, we prove that there is a β 0 >0 such that if β < β 0 , 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.