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, under the ‐norm constraint where , , and 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.