EXISTENCE AND MULTIPLICITY OF HOMOCLINIC ORBITS FOR POTENTIALS ON UNBOUNDED-DOMAINS

We study the system q = - V'(q) in R(N), where Y is a potential with a strict local maximum at 0 and possibly with a singularity. First, usi ng a minimising argument, we can prove the existence of a homoclinic o rbit when the component OMEGA of {x is-an-element-of R(N): V(x) < V(0) } containing 0 is an arbitrary open set; in the case OMEGA unbounded w e allow V(x) to go to 0 at infinity, although at a slow enough rate. T hen we show that the presence of a singularity in OMEGA implies that a homoclinic solution can also be found via a minimax procedure and, co mparing the critical levels of the functional associated to the system , we see that the two solutions are distinct whenever the singularity is 'not too far' from 0.