Existence Of Homoclinic Orbits For Second Order Hamiltonian Systems

In this thesis, we investigate the existence of homoclinic orbits for severalkinds of second order Hamiltonian systems and ordinary p-Laplacian systems byusing the variational methods and critical point theory. The thesis is divided into fivechapters, which is organized as follows:In chapter1, we introduce the historical background and the recentdevelopment of the problems considered in this thesis, as well as the main workdone in this dissertation. Moreover, we present some preliminary lemmas anddefinitions.In chapter2, we study the existence of homoclinic orbits for second orderHamiltonian systems, where the potential is periodic. We mainly consider two casesfor the above systems, including that the potential is respectively superquadratic andnon-coercive. At first we prove the existence of homoclinic orbits for a kind ofsuperquadratic periodic Hamiltonian systems via Mountain Pass theorem, and thensome sufficient conditions for guaranteeing the existence of a kind of non-coerciveperiodic Hamiltonian systems are given by using the extreme value principle.In chapter3, we discuss the existence of homoclinic orbits for second ordernonperiodic Hamiltonian systems, where the potential satisfies superquadratic orsome general local conditions, respectively. Under the above two cases, someexistence of homoclinic orbits are obtained by using critical point theory andtechnique of localization.In chapter4, we consider the existence of homoclinic orbits for ordinaryp-Laplacian systems. We establish some sufficient conditions for the existence ofhomoclinic orbits for the above p-Laplacian systems, which generalize and improvethe results in chapter2. In chapter5, we summarize the main results in this thesis and propose someproblem which can be considered in the future study.