Normalized solutions and asymptotical behavior of minimizer for the coupled Hartree equations

We prove the existence and nonexistence of L2(RN)-normalized solutions of coupled Hartree equations, which arisen from the studies of the nonlinear optics and multi-component Bose-Einstein condensates. Under certain type trapping potentials, by proving some delicate energy estimates, we give a precise description on the concentration behavior of minimizer solutions of the system. Furthermore, an optimal blowing up rate for the minimizer solutions of the system is also proved.