Homoclinic Orbits For Hamiltonian Systems Without Periodicity

Hamiltonian system theory is both classical and modern research area, which is studied by different methods. In this paper, we use variational method. The solutions of Hamiltonian systems will be obtained as critical points of the corresponding functionals. During the past twenty years there has been a great deal of progress in the use of variational method to find periodic, homoclinic, heteroclinic orbits for Hamiltonian systems.In this paper, the existence of nontrivial homoclinic orbits of Hamiltonian systems without periodicity and with potentials changing signis proved, where is a potential changing sign. Assume L(t) and V(t,u) satisfy is a positive definite symmetric matrics, there exists an suth thatuniformaly for(V₂) there exist constants μ>2,1≤β<2,r> 0,d₁≥ 0 suth that(V₃) there exists a function V₁(u)∈C(?)suth thatUnder the assumptions (V₁)-(V₃), there exists d₂≥0 suth thatIf V(t,u)also satisfy (V₄) there exists a pair (t₀,u₀) suth that |u_o|-1 and V(t_o,u_o)>d₂/(μ-β)Then (HS2) possesses at least one nontrivial homoclinic orbit. Under the above assumptions, if V(t,u) satisfy...