Marek Izydorek

Gdansk University of Technology (Poland)

Generalized 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

q+ Vq(t,q) = 0,
(HS)

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:

  1. #M 2 and γ := inf{|x - y|: x,y ∈ M, x ⁄= y} > 0  ,
  2. there exists 0 < ɛ0 G such that for every 0 < ɛ ɛ0 there is δ > 0 such that for all (t,x) ∈ R×Rn if d(x,M) ɛ then -V (t,x) > δ,
  3. for every x ∈ Rn \ M, lim      - V (t,x) = + ∞
--t→ ∞ ,
  4. for every x ∈ M, ∫             √----
-+∞∞ - V (t,x) < 2α0  , where
    α  := inf{- V (t,x): d(x,M ) ≥ ɛ }.
 0                         0

Our result states that each point of M is joined with a certain other element of M by a solution of (HS). Since we should not expect that (HS) possesses a stationary solution the notion of a heteroclinic orbit is used in a generalized sense. Namely, q: R Rn is a generalized heteroclinic solution of (HS) if there exist x,y ∈ Rn, xy such that q joins x to y, (i.e. limt→-∞q(t) = x, limt+q(t) = y).