Existence and stability of standing waves for nonlinear fractional Schrödinger equations

In this paper, we consider the nonlinear fractional Schrödinger equations. By studying constrained minimization problems and applying the method of concentration-compactness and commutator estimates, we obtain the existence of standing waves for nonlinear fractional Schrödinger equations under some assumptions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, in the sense that a solution which starts near the set will remain near it for all time.