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 − (φ(|ú|)ú)́ + V ́(u) = 0 in R, where V is a double-well potential with minima at t = ±α and φ : (0, +∞) → (0, +∞) is a C1 function satisfying some technical assumptions. Our results include the classic case φ(t) = tp−2, which is related to the celebrated p-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case φ(t) = √1+1t2 , which is directly associated with the prescribed mean curvature operator.