LAUSR

Abstract We prove the existence of monotone heteroclinic solutions to a scalar equation of the kind u ″ = a ( t ) V ′ ( u ) under the following assumptions: V ∈ C 2 ( R ) is a non-negative double well potential which admits just one critical point between the two wells, a ( t ) is measurable, asymptotically periodic and such that inf ⁡ a > 0 , sup ⁡ a + ∞ . In particular, we improve earlier results in the so called asymptotically autonomous case, when the periodic part of a, say a ˜ , is constant, i.e. a ( t ) converges to a positive value l as | t | → + ∞ . Furthermore, whenever a ˜ fulfils a suitable non-degeneracy condition, the solutions are shown to be infinitely many.