Existence and stability of standing waves for nonlinear fractional   Schr\"odinger equations with Hartree type nonlinearity
  In this paper, we consider the nonlinear fractional Schr\"odinger equations with Hartree type nonlinearity. We obtain the existence of standing waves by studying the related constrained minimization problems by applying the concentration-compactness principle. Via symmetric decreasing rearrangements, we also show that the standing waves, up to a translations and phases, are positive symmetric nonincreasing functions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near it for all time. 
Dan Wu
math.AP