带有共振的二阶哈密顿系统非平凡解的存在性
The Existence of Nontrivial Solutions to Second Order Hamiltonian Systems Which Are Resonant at Infinity

本文研究二阶哈密顿系统的非平凡解问题. 假设系统中的非线性项$V'$是渐近线性的. 利用变分法, 通过系统对应泛函的小扰动的临界点来建立系统的Palais-Smale序列, 进而说明该序列的有界性. 与一般做法不同的是, 本文对$V'$不限定Landesman-Lazer条件.
In this paper, the author considers nontrivial solutions to the second\linebreak order Hamiltonian system. Nontrivial solutions are obtained under the assumption that the asymptotically linear nonlinearity $V'$ is resonant at infinity. The arguments are variational. The author constructs the Palais-Smale sequence from a sequence of exact critical points of nearby functionals, possessing extra properties which help to insure its boundness. Different from the existing results in the literature, we do not make any Landesman-Lazer resonance conditions on $V'$.