无穷远处渐近二次增长的哈密顿系统的新同宿轨道
New homoclinic orbits for Hamiltonian systems with asymptotically quadratic growth at infinity
摘要：
本文研究了一类在无穷远处具有渐近二次非线性项的哈密顿系统\begin{eqnarray*} \ddot{u}（t）-L（t）u+ \nabla W（t，u）=0的同宿解的存在性和多解性. {eqnarray*}我们引入了一个新的强制条件，得到了一个新的嵌入定理。在此基础上，利用广义山路定理证明了上述系统至少有一个非平凡同宿轨道。利用变量喷泉定理，在对称条件下得到了该问题的无穷多个同宿轨道。我们的渐近二次性条件与文献中已有的条件不同。
英文摘要：
In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity \begin{eqnarray*} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0. {eqnarray*} We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by Generalized Mountain Pass Theorem. By Variant Fountain Theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.
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