Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Existence and multiplicity of homoclinic orbits for potentials on unbounded domains

Paolo Caldirolia1

a1 SISSA, via Beirut 2–4, 34014 Trieste, Italy

Abstract

We study the system in RN, where V is a potential with a strict local maximum at 0 and possibly with a singularity. First, using a minimising argument, we can prove the existence of a homoclinic orbit when the component Ω of {x ∈ RN: V(x) < V(0)} containing 0 is an arbitrary open set; in the case Ω unbounded we allow V(x) to go to 0 at infinity, although at a slow enough rate. Then we show that the presence of a singularity in Ω implies that a homoclinic solution can also be found via a minimax procedure and, comparing 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.

(Received October 30 1992)

(Revised March 31 1993)