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| In this paper, we consider nonlinear pseudo-relativistic Schr\"odinger equations with Hartree type nonlinearities.We study their standing waves based on a variational framework.Instead of using a scaling argument, we employ coupled rearrangement inequalities to exclude the lack of compactness in the constrained minimizing problems,due to the inhomogeneous and nonlocal terms in the equations.To overcome these difficulties,we establish two new coupled rearrangement inequalities according to fully nonlinear pseudo-relativisticoperators and Hartree type nonlinearities.As a consequence, the existence of standing wavesis obtained via applying the concentration-compactness principle in subcritical case.Moreover, we also derive symmetric results and other properties of the minimizers. |
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| Keywords:Schr\"odinger equation; Variational framework; The constrained minimizing problems; Coupled rearrangement inequalities; The concentration-compactness principle. |
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