  In this paper, we prove the existence, uniqueness and qualitative properties
of heteroclinic solution for a class of autonomous quasilinear ordinary
differential equations of the Allen-Cahn type given by
  $$
  -\left(\phi(|u'|)u'\right)'+V'(u)=0~~\text{ in }~~\mathbb{R},
  $$
  where $V$ is a double-well potential with minima at $t=\pm\alpha$ and
$\phi:(0,+\infty)\to(0,+\infty)$ is a $C^1$ function satisfying some technical
assumptions. Our results include the classic case $\phi(t)=t^{p-2}$, which is
related to the celebrated $p$-Laplacian operator, presenting the explicit
solution in this specific scenario. Moreover, we also study the case
$\phi(t)=\frac{1}{\sqrt{1+t^2}}$, which is directly associated with the
prescribed mean curvature operator.
