Heteroclinic solutions of singular quasilinear bistable equations

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In this note we consider the action functional $$\begin{aligned} \int _{\mathbb {R}\times \omega } \left( 1 - \sqrt{1 - |\nabla u|^{2}} + W(u) \right) \mathrm {d}\bar{x} \end{aligned}$$ where W is a double well potential and \(\omega \) is a bounded domain of \(\mathbb {R}^{N-1}\). 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 heteroclinic connection for the non-autonomous model $$\begin{aligned} \int _{\mathbb {R}} \left( 1 - \sqrt{1 - |u'|^{2}} + a(t)W(u) \right) \mathrm {d}t \end{aligned}$$ is also addressed. For this functional, we look for the possible assumptions on a(t) ensuring the existence of a minimizer.