Ground State Solutions And Periodic Solutions With Minimal Periods To Second-order Hamiltonian Systems

This paper mainly studies the existence of ground state solutions and minimum periodic solutions for autonomous and non-autonomous second-order Hamiltonian systems by applying variational methods and Nehari manifold methods.Specifically,Chapter 2 investigates the existence of ground state solutions and minimal periodic solutions for non-autonomous second-order Hamiltonian systemü(t)+A(t)u(t)+▽H(t,u(t))=0,t ∈ R.Among them,the nonlinear term has a special form,which is even with respect to the variable u and satisfies the monotonic non-decreasing assumption.By using the variational method,the existence of periodic solutions of the system is transformed into the existence of critical points in a subspace of the corresponding variational functional space of the system.By establishing the homeomorphism mapping between the Nehari manifold and the unit sphere on the subspace,we can find the functional critical point on the Nehari manifold,which corresponds to the minimum periodic solution of the system,and obtain some sufficient conditions to ensure the existence of the minimum periodic solution of the Hamiltonian system.The results of this article generalize the results in existing literature.Chapter 3 considers the existence of periodic solutions for autonomous secondorder Hamiltonian system x+V’(x)=0,x∈RN,where V is even.Through the variational method,we transform the existence of periodic solutions of the Hamiltonian system into the existence of critical points in a subspace of the corresponding functional of the system.We adopt the monotonicity assumption introduced by Bartsch and Mederski in[1]and remove the strict convexity assumption.Then,in this case,we can establish the homeomorphic mapping between the Nehari manifold and the unit sphere in a subspace of the space where the functional is located.By using the Nehari manifold method introduced by Szulkin(see[2]),we prove the existence of the minimal periodic solution of the system.