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In this paper, we consider the nonlinear fractional Schr\"odinger equationswith Hartree type nonlinearity. We obtain the existence of standing waves bystudying the related constrained minimization problems by applying theconcentration-compactness principle. Via symmetric decreasing rearrangements,we also show that the standing waves, up to a translations and phases, arepositive symmetric nonincreasing functions. Moreover, we prove that the set ofminimizers is a stable set for the initial value problem of the equations, thatis, a solution whose initial data is near the set will remain near it for alltime.

Existence and stability of standing waves for nonlinear fractional Schr\"odinger equations with Hartree type nonlinearity