Normalized solutions to nonlinear Schrödinger equations with competing Hartree-type nonlinearities

In this paper, we consider solutions to the following nonlinear Schrödinger equation with competing Hartree-type nonlinearities, $$-\mathrm{\Delta }u+\lambda u=(\mathrm{\mid }x{\mathrm{\mid }}^{-{\gamma }_1}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u-(\mathrm{\mid }x{\mathrm{\mid }}^{-{\gamma }_2}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)uin{\mathbb{R}}^N,$$ under the $L^2$-norm constraint $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=c>0,$$ where $N\ge 1$, $0<{\gamma }_2<{\gamma }_1<\min \{N,4\}$ and $\lambda \in \mathbb{R}$ appearing as Lagrange multiplier is unknown. First we establish the existence of ground states in the mass subcritical, critical and supercritical cases. Then we consider the well-posedness and dynamical behaviors of solutions to the Cauchy problem for the associated time-dependent equations.