On Minimizers of the Hamiltonian System u" = delW (u) and on the Existence of Heteroclinic, Homoclinic and Periodic Orbits

In the first part of the paper, we establish two necessary conditions for the existence of bounded one-dimensional minimizers u: the potential W must have a global minimum supposed to be 0 without loss of generality, and W(u(x)) -> 0 as vertical bar x vertical bar -> infinity. Furthermore, non-constant minimizers connect at 00 two distinct components of the set {W = 0}. In the second part, we prove (when the previous assumptions are satisfied) the existence of nontrivial minimizers. We also show the existence of heteroclinic, homoclinic, and periodic orbits in analogy with the scalar case. Finally, we study the asymptotic convergence of these solutions.