Heteroclinic connections for a double-well potential with an asymptotically periodic coefficient

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∈C2(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.