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arXiv:2404.11693v1 Announce Type: new
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
$$
-\left(\phi(|u'|)u'\right)'+V'(u)=0~~\text{ in }~~\mathbb{R},
$$
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.
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