Coupled Rearrangement Inequalities And Their Applications For Schr(?)dinger Equations

With the development of science and technology,partial differential equations are gradually integrated into our life and work due to their own advantages.At the same time,more and more mathematicians begin to pay attention to the nonlinear Schr(?)dinger equations.In this paper,we consider the nonlinear pseudo-relativistic Schr(?)dinger equations with Hartree type nonlinearities.We study their standing waves based on a variational framework.On the one hand,we prove several important rearrangement inequalities,which provide a strong theoretical basis for further research on their applications in nonlinear pseudorelativistic Schr(?)dinger equations.Firstly,we summarize the known results such as concepts and properties of function spaces,rearrangement functions and coupled rearrangement functions.We also give the related basic inequalities and important theorems.Secondly,we prove the rearrangement inequalities and coupled rearrangement inequalities for Hartree nonlinear terms,fractional Laplacian operators and general pseudo-relativistic Schr(?)dinger operators.Thirdly,we deduce the energy rearrangement inequality and the strict energy coupled rearrangement inequality.Moreover,it is shown that these inequalities provide a foundation for proving the subadditivity conditions of the energy and verifying the existence and properties of the ground states.On the other hand,we illustrate the applications of coupled rearrangement inequalities in nonlinear pseudo-relativistic Schr(?)dinger equations.Firstly,we show some properties of the constrained minimization problems,especially by deducing the subadditivity conditions.In general,scaling arguments are used to obtain subadditivity conditions which play a key role in excluding lack of compactness in constraint minimization problems.Due to the inhomogeneous and nonlocal terms in the equations considered in this paper,we cannot use a scaling argument,but instead use two new coupled rearrangement inequalities to exclude the lack of compactness in the constrained minimization problem.Secondly,in the subcritical case,we obtain the existence of ground states for the general pseudo-relativistic Schr(?)dinger equations with Hartree type nonlinearities.Finally,in the case of p = 2,we show the compactness and orbital stability of solutions to the nonlinear pseudo-relativistic Schr(?)dinger equations by applying the concentration-compactness principle.