Izydorek-Janczewska超二次非周期哈密顿系统同宿轨道解存在定理的一个新证明

A New Proof on Existence of Homoclinic Orbits for a Class of Superquadratic Aperiodic Hamiltonian Systems

本文考虑超二次非周期哈密顿系统q+V_(q)(t,q)=f(t),t∈R(HS)同宿轨道解的存在性,其中V(t,q)=-K(t,q)+W(t,q)关于变量t是T-周期的,且满足条件(V_(1))-(V_(7)),f∈L^(∞)(R,R^(n))∩L^(2)(R,R^(n)),虽然相应的能量泛函φ不满足(PS)条件,但可利用Brezis-Nirenberg型的山路定理得到φ的一个(PSC)c*序列{q_(m)},进而可证明{q_(m)}弱收敛到(HS)的一个非平凡同宿轨道解。

In this paper, we consider the existence of homoclinic orbits for the nonperiodic superquadratic Hamiltonian system q+V_(q)(t,q)=f(t),t∈R(HS) where V(t,q)=-K(t,q)+W(t,q) is T-periodic in t,and satisfies the condition(V_(1))-(V_(7)),f∈L^(∞)(R,R^(n))∩L^(2)(R,R^(n)).Although the corresponding energy functional φ does not satisfy the(PS)condition, a (PSC)c* sequence{q_(m}can be obtained by Brezis-Nirenberg type mountain pass theorem, and then we can prove that {q_(m)} weakly converges to a nontrivial homoclinic orbit of(HS).