Heteroclinic solutions of singular quasilinear bistable equations

In this note we consider the action functional integral(Rx omega) (1-root 1-vertical bar del u vertical bar(2) + W(u))d (x) over bar where 147 is a double well potential and is a bounded domain of RN-1. We prove existence, one-dimensionality and uniqueness (up to translations) of a smooth minimizing phase transition between the two stable states u = 1 and u 1. The question of existence of at least one minimal heteroctinic corniection far the non-autonomous model integral(R) (1-root 1-vertical bar u'vertical bar(2+)a(l)W(u))dl is also addressed. For this functional, we look for the possible assumptions on a(t) ensuring the existence of a minimizer.