Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity

Abstract

In this paper, we consider the nonlinear fractional Schrödinger 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.

References in corpus (4)

• Fractional Quantum Mechanics
• Profile decompositions and Blowup phenomena of mass critical fractional Schrödinger equations
• On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity
• Some Fractional Functional Inequalities and Applications to Some Constrained Minimization Problems Involving a Local Non-linearity