Existence And Multiplicity Of Periodic Solutions And Homoclinic Orbits For Several Kinds Of Differential Systems

By applying the least action principle, mountain pass lemma?linking theo-rem and fountain theorem in critical point theory, this doctoral dissertation mainly deals with the existence and multiplicity of periodic solutions and homoclinic so-lutions for Hamiltonian systems and p-Laplace systems. This thesis is made up by four sections.In Chapter 1, a brief introduction is given to the historic background, status and the up-to-date progress for all the investigated problems. Some preliminaries are also introduced in this chapter.In Chapter 2, by using the least action principle and saddle point theorem, the existence of periodic solutions for the following second order Hamiltonian system is investigated. Some sufficient conditions are obtained which generalize and im-prove some known results. By using mountain pass lemma and linking theorem, the existence of periodic solutions of the second order Hamiltonian system is studied. Some new theorems are obtained under the local superquadratic con-dition and other conditions.In Chapter 3, the existence of homoclinic solutions for the following Hamilto-nian system is considered. When W is superquadratic at infinity, the above Hamiltonian system has at least one homoclinic solution by using the variant mountain pass lemma; when W is subquadratic at infinity, there are infinitely many homoclinic solutions for the above Hamiltonian system by applying fountain theorem.In Chapter 4, the existence of periodic solutions and homoclinic solutions for the following p-Laplace system is studied. Infinitely many periodic solutions are obtained for the p-Laplace system The results obtained generalize those in some known literatures.