On homoclinic and heteroclinic orbits for Hamiltonian systems

We extend some earlier results on existence of homoclinic solutions for a class of Hamiltonian systems.We also study heteroclinic solutions.We use a variational approach. Introduction.Recently variational techniques have been used in a number of papers to obtain existence of homoclinic and heteroclinic orbits of the Hamiltonian systems u 00 L(t)u + V u (t, u) = 0;(1.1) see, e.g., A. Ambrosetti and M.L. Bertotti ([1]), P.H. Rabinowitz ([7]), W. Omana and M. Willem ([5]), and P. Korman and A.C. Lazer ([3]).Here L(t) is a given positive definite n ⇥ n matrix, the potential V (t, u) is assumed to be superquadratic in u, and the solution is sought in the class H 1 (R, R n ), which implies that it is homoclinic at zero; i.e., lim t!±1 u(t) = 0.The approach used in [1], [5] and [3], was to restrict the problem (1.1) to a bounded interval ( T, T ) with Dirichlet boundary conditions u( T ) = u(T ) = 0, show existence of solutions using the mountain-pass lemma, and then let T ! 1.The crucial observation made in [1], and independently in [3], is that in addition to existence of solutions, the mountainpass lemma allows one to obtain a uniform-in-T estimate of H 1 norm of the solution.It is then straightforward, via the usual diagonal process, to show existence of a homoclinic solution of (1.1).The problem is to show that this solution is nontrivial.P.H. Rabinowitz and K. Tanaka proved existence of a solution under the condition that the smallest eigenvalue of L(t) tends to 1 as |t| !1; see [8], and also [5], where