Hamiltonian System With The Solution Of The Schrodinger Equation

With the development of science and technology, various nonlinear prob-lem has aroused people's widespread interest day by day, the nonlinear analysis has become one of the important research directions of modern mathematics. It mainly studies all kinds of nonlinear differential equations, the variational ap-proach is one of its important study methods. Variational method to differential equations is a method that transfer boundary value problem of differential equa-tions into variational problem in order to prove existence of solutions, determine the number of solutions, and seek its approximate solution. Hamiltonian systems and Schrodinger equations are two common problems, which can be solved by variational methods. In this paper, we use the generalled mountain pass theo-rem, local linking theorem and fountain theorem to study some boundary value problem of non-linear differential equations.The thesis is divided into two chapters according to contents.In chapter 1, we use the genrealled mountain pass theorem, local linking theorem and fountain theroem to investigate the periodic solutions of a class of nonautonomous second order Hamiltonian systems respectively, where T> 0, B∈C(R,RN2) is a symmetric matrix-valued function with (B(t)x,x)≥0 for all x∈RN and all t∈R. H(t,x) be measurable in t for each x∈RN and continuously differentiable in x for almost every t∈R, H is T-periodic in t and H(t,0)= 0, after giving more assumptions, we obtain some existence and multiplicity theorems of periodic solutions for this class of nonautonomous second order Hamiltonian systems.In chapter 2, firstly, we use fountain theorem to investigate the existence of infinitely solutions for the superlinear Schrodinger equation where V E C(RN, R)satisfies infx∈RN V(x)≥a1> 0, a1> 0 is constant. There is a r, such that for all b> 0, lim|y|â†'∞meas({x∈RN|V(x)≤b}∩Br(y))= 0. After giving more assumptions, we obtain some theorems about the existence of infinitely solutions for the superlinear Schrodinger equation(2.1). Secondly, we use fountain theorem to investigate the existence of infinitely solutions for the superlinear superlinear Schrodinger-Maxwell equation, At the same assumptions on Vas in (2.1), we obtain some theorems about the existence of infinitely solutions for the superlinear Schrodinger-Maxwell equa-tion(2.2).