The Existence And Stability Of Solutions For Nonlinear Fractional Schr(?)dinger Equations

In this paper,we mainly concern with the existence and stability of solutions for nonlinear fractional Schr(?)dinger equations.At first,we prove the existence and uniqueness of the global solution for the generalized nonlinear fractional Schr(?)dinger equations with periodic boundary conditions.Then we consider the existence and stability of standing waves for a class of the nonlinear fractional Schr(?)dinger equations with Hartree type and power type nonlinearities.At last,we prove the existence and stability of standing waves for a class of the nonlinear fractional Schr(?)dinger equation with combined nonlinearities of multiple potential and Hartree type.This paper is organized in four chapters.In the first part,we give the physical background,the research status of Schr(?)dinger equations and Hartree equations.We recall some known results and briefl y describe the main content of this article.In the second part,the periodic boundary condition of the generalized nonlinear fractional Schr(?)dinger equations is studied.We obtain the priori estimates by using H(?)lder inequality,Gagliardo-Nirenberg inequality,Young inequality and Gronwall inequality.Then we construct the global approximate solution by the Gal?rkin method,and prove the existence and uniqueness of the global solution for the generalized nonlinear fractional Schr(?)dinger equations.In the third part,we consider the existence and stability of standing waves for nonlinear fractional Schr(?)dinger equations with Hartree type and power type nonlinearities.By studying variational problem,the concentration-compactness principle and commutator estimates,we obtain the existence and stability of standing waves.In the fourth part,we prove the existence and stability of standing waves for nonlinear fractional Schr(?)dinger equations with combined nonlinearities of multiple potential and Hartree type.We obtain the existence of standing waves by studying the constrained minimization problem and applying the concentration-compactness principle.Furthermore,we prove the stability of the standing waves.