Existence and Stability of Standing Waves for Nonlinear Fractional Schrödinger Equations with Hartree Type and Power Type Nonlinearities

We consider the standing wave solutions for nonlinear fractional Schrödinger equations with focusing Hartree type and power type nonlinearities. We first establish the constrained minimization problem via applying variational method. Under certain conditions, we then show the existence of standing waves. Finally, we prove that the set of minimizers for the initial value problem of this minimization problem is stable