Uniqueness of Heteroclinic Solutions in a Class of Autonomous Quasilinear ODE Problems

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 [Formula: see text] where [Formula: see text] is a double-well potential with minima at [Formula: see text] and [Formula: see text] is a [Formula: see text] function satisfying some technical assumptions. Our results include the classic case [Formula: see text], which is related to the celebrated [Formula: see text]-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case [Formula: see text], which is directly associated with the prescribed mean curvature operator.