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\u2033=a(t)V\u2032(u) under the following assumptions: V 08C2(R) is a non-negative double well potential which admits just one critical point between the two wells, a(t)a(t) is measurable, asymptotically periodic and such that inf_a>0, sup_a<+ 1e. In particular, we improve earlier results in the so called asymptotically autonomous case, when the periodic part of a, say \alpha, is constant, i.e. a(t) converges to a positive value l as |t|\u2192+ 1e. Furthermore, whenever \alpha fulfils a suitable non-degeneracy condition, the solutions are shown to be infinitely many