Research On The Existence Of Solutions For Hamiltonian Systems And Schr(?)dinger-Poisson Systems

In this Ph.D. thesis, we deal with the existence of periodic solutions and homoclinic orbits for several types of nonlinear Hamiltonian systems and the existence of solutions for two types of nonlinear Schrodinger-Poisson systems by using variational methods, and obtain some new the existence and multiplicity results. The dissertation is divided into five chapters. The main contents are as follows:In chapter 1, the historical background, status and the up-to-date progress for all the investigated problems are introduced, the main contents of the dissertation are outlined, and at the end of this chapter, some preliminary tools used in proving the main results are given.In chapter 2, using the variational methods for strong indefinite problems, we prove the existence and multiplicity of homoclinic obits for the following first order Hamiltonian systems with spectrum point zero: where J is the standard symplectic matrix in R2N: and H C1(R×R2N, R) has the form with L being a 2N×2N symmetric constant matrix, and W?C1(R×R2N,R). We consider two cases for the above systems as follows:(1) H(t, u) is 1-periodic in t, and W(t, u) is asymptotically quadratic and superquadratic as |u|??, respectively;(2) H(t,u) is nonperiodic in t, and W(t,u) is asymptotically quadratic as |u|??.In chapter 3, we study the existence of infinitely many homoclinic orbits for the following second order Hamiltonian systems with the subquadratic potential W where W?C1(R, RN), and L?C(R, RN×N) is a symmetric matrix valued function. We solve an open problem in the literature, and obtain a sufficient condition for the existence of infinitely many homoclinic orbits for the above systems. The result is generalized and improved in the literature.In chapter 4, we discuss the existence and multiplicity of periodic solutions for the following two types of second order Hamiltonian systems with impulsive effects: and and a class of second order differential equations with impulsive effects: By using some recent critical points theorems, we obtain some existence and multiplic-ity results for the above three systems which extend and improve some results in the literature.In chapter 5, using the modified version of the mountain pass theorem, we study the existence of solutions for the following Schrodinger-Poisson system: We consider two cases for the above elliptic system as follows:(1) V,a:RN?R+ are radial and smooth, K is a positive constant, and f is asymptot-ically linear as |u|??;(2) V=1, K?L2(R3,R+), a?C(R3,R+), and f is asymptotically linear as |u|??.Under the above two cases, we obtain the existence of a positive solution and a ground state solution, respectively. Some recent results are extended and supplied.