一类超二次哈密顿系统的无穷多周期解

Periodic Solutions of Second-order Hamiltonian Systems with Super-quadratic Potentials

研究具有超二次势能的二阶Hamilton系统ü+A(t)u(t)+F(t,u(t))=0,u(0)-u(T)=.u(0)-.u(T)=0无穷多周期解的存在性问题.在线性项非零的假设下,当位势函数F满足新的超二次条件而不满足Ambrosetti-Rabi-nowitz条件时,运用临界点理论中喷泉定理证明此系统存在无穷多非平凡的周期解.

We study the existence of infinitely many periodic solutions for some second-order Hamiltonian systems ： {ü＋A（t）u（t）＋▽F（t,u（t））=0,u（0）-u（T）=ù（0）-ù（T）=0, which are with super-quadratic potentials. Under the assumption that the linear part is not equivalent to O, the existence of infinitely many nontrivial periodic solutions for the systems is proved with the variant fountain theorem in critical point theory, where F（t,u） satisfies a new super-quadratic condition and need not satisfy the global Ambrosetti-Rabinowitz condition.