Heteroclinics for non-autonomous second-order differential equations

We investigate new conditions for the existence of heteroclinic solutions of a non-autonomous equation of the form u''=a(t)f(u), where a(t) is a bounded, positive function, f(-1)=f(1)=0, and f=F', where F is a C^1, non-negative function such that F(-1)=F(1)=0. We are mainly interested in the case where a(t) approaches its positive limit from above, as |t| diverges, but we allow also the "asymptotically asymmetric" case, where the difference between the two limits (at minus infinity and plus infinity) is a sufficiently small positive number. Variational methods are used in the proof.