On a class of infinite-dimensional Hamiltonian systems with asymptotically periodic nonlinearities

Abstract

The authors study the existence of homoclinic type solutions for the following system of diffusion equations on ℝ × ℝ ^N$\left\{ \begin{gathered} \partial _t u - \Delta _x u + b \cdot \Delta _x u + au + V(t,x)v = H_v (t,x,u,v), \hfill \\ - \partial _t v - \Delta _x v - b \cdot \Delta _x v + av + V(t,x)u = H_u (t,x,u,v), \hfill \\ \end{gathered} \right.$ where z = (u, v): ℝ × ℝ ^N → ℝ ^m × ℝ ^m, a > 0, b = (b ₁, …, b _N) is a constant vector and V ε C(ℝ × ℝ ^N, ℝ), H ε C ¹ (ℝ × ℝ ^N × ℝ^2m, ℝ). Under suitable conditions on V(t,x) and the nonlinearity for H(t, x, z), at least one non-stationary homoclinic solution with least energy is obtained.

Keywords

Variational methods / Least energy solution / Hamiltonian system

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