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

Abstract

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 $$-{\left(\varphi (|u^{\prime }|)u^{\prime }\right)}^{\prime }+V^{\prime }(u)=0\text{ in }\mathbb{R},$$ where $V$ is a double-well potential with minima at $t=\pm \alpha$ and $\varphi :(0,+\infty )\to (0,+\infty )$ is a $C^1$ function satisfying some technical assumptions. Our results include the classic case $\varphi (t)=t^{p-2}$, which is related to the celebrated $p$-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case $\varphi (t)=\frac{1}{\sqrt{1+t^2}}$, which is directly associated with the prescribed mean curvature operator.