Application Of The Cerami Condition Critical Point Theorem

Nonlinear problems in mathematics originated from physics, chemistry, biol-ogy, celestial mechanics, economic and social science. It is manifested in the form of a variety of nonlinear equations, mainly non-linear differential equation.The branch of the nonlinear mathematical problems is called nonlinear analysis. To study nonlinear differential equations,it established a set of systems and deep mathematical theory. In this paper, as some applications of some abstract critical point theorems under Cerami condition, we study the existence and multiplicity of solutions for Hamiltonian systems.The thesis is divided into three chapters according to contents:In chapter 1, It is concerned with the following second order Hamiltonian systems: Under a general assumption, the existence of infinitely many periodic solutions are obtained by the variant fountain theorem under Cerami condition.The superquadratic condition in our situation is far weaker than the classic one of Ambrosetti-Rabinowitz (AR)([1]): It is also weaker than the current commonly used superquadratic condition.In chapter 2, we consider the following first order infinite-dimensional Hamil-tonian systems: we obtain one abstract result on existence of infinitely many critical point by means of critical points theorem for strongly indefinite symmetrical function un-der Cerami condition. At the same time,our results generalize many recent stud- In chapter 3,we consider the existence of subharmonic solutions of the su-perquadratic Hamiltonian system(2.1.1). The existence of infinitely many sub-harmonic solutions for the system is proved with Generalized Mountain Pass Theorem under Cerami condition.