Two-end solutions to the Allen-Cahn equation in ${\mathbb{R}}^3$

Changfeng Gui, Yong Liu, Juncheng Wei

In this paper we consider the Allen-Cahn equation $$ -Δu = u-u^3 \ \mbox{in} \ {\mathbb R}^3 $$ We prove that for each $k\in \left(\sqrt{2},+\infty \right),$ there exists a solution…