Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems

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We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in ${\Bbb R}^N$ of the form $\ddot q=q-W'(t,q)$ , where we assume the existence of a sequence $(t_n)\subset{\Bbb R}$ such that $t_n\to \pm\infty$ and $W'(t+t_n,x)\to W'(t,x)$ as $n\to\pm\infty$ for any $(t,x)\in{\Bbb R}\times{\Bbb R}^N$ . Moreover, under a suitable non degeneracy condition, we prove that this class of systems admits multibump solutions.

Received February 2, 1996 / In revised form July 5, 1996 / Accepted October 10, 1996