Homoclinic solutions of an infinite-dimensional Hamiltonian system

Abstract.

We consider the system

\begin{equation*} \left\{ \begin{array}{l} \partial_t u-\Delta_x u+V(x)u = H_v(t,x,u,v) [3mm] -\partial_t v-\Delta_x v+V(x)v = H_u(t,x,u,v) \end{array}\qquad\text{for }(t,x)\in{\mathbb R}\times{\mathbb R}^N \right. \end{equation*}

which is an unbounded Hamiltonian system in $L^2({\mathbb R}^{N},{\mathbb R}^{2M})$. We assume that the constant function $(u_0,v_0)\equiv(0,0)\in{\mathbb R}^{2M}$ is a stationary solution, and that H and V are periodic in the t and x variables. We present a variational formulation in order to obtain homoclinic solutions z=(u,v) satisfying $z(t,x)\to 0$ as $|t|+|x|\to\infty$. It is allowed that V changes sign and that $-\Delta+V$ has essential spectrum below (and above) 0. We also treat the case of a bounded domain $\Omega$ instead of ${mathbb R}^N$ with Dirichlet boundary conditions.