Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C1(R×Rn,R), V⩽0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M⩾2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)⩾ε then −V(t,z)⩾δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn∖M. Our result states that each point in M is joined to another point in M by a solution of our system.