Ground State Solutions Of Hamiltonian Systems

With the continuous development of science and technology, variety of nonlin-ear problems are becoming increasingly attracted people’s attention. The nonlinear analysis has become one of the important research directions in modern mathematic, A variety of nonlinear differential equation is one of the main objects for study, The variational method is one of the important research methods for nonlinear analysis. Variational methods in differential equations is transform the boundary value problems into variational problems to prove the existence of solutions, for example, the Hamilto-nian system and the Schrodinger equation are two common variational method to solve the problem. In this paper, we use Nehari manifold, weak link theorem, the minimax principle and Ekelands variational principle to study several types of super-secondary non-linear differential equation boundary value problem.Based on the inherent relationship between the structure and content of study, this paper include the following chapters:Chapter1In this chapter, we use the Ekelands variational principle discuss the existence of a class of super-quadratic Hamiltonian systems’s ground state solution, and we use the minimax principle consider the multiplicity of Solutions. We consider the following equation where A(t) is a continuous T—periodic symmetric matrix, H:R x RNâ†'R is T-periodic (T>0), we always assume that H(t,x) is continuous in t, for each x∈RN, continuously differentite in x, for each t∈[0,T], andâ–½H(t,x) denotes its gradient with respect to the x variable.Chapter2In this chapter, we use the promoted weak link theorem study the following equation H:R×RNâ†'R is T—periodic (T>0) in its first variable. Moreover we assume that H(t,x) is continuous in t, for each x∈RN, continuously differentiable in x, for each t∈E [0,T], and we useâ–½H(t,x) denotes its gradient with respect to the x variable.