Qualitative properties of two-end solutions to the Allen–Cahn equation in R3 ${\mathbb{R}}^{3}$

A solution of the Allen–Cahn equation in R 3 ${\mathbb{R}}^{3}$ is called a two-end solution if its nodal set is asymptotic to ( x ′ , z ) ∈ R 3 : z = k i ln | x ′ | + c i , 1 ≤ i ≤ 2 $\left\{\left({x}^{\prime },z\right)\in {\mathbb{R}}^{3}:z={k}_{i}\mathrm{ln}\vert {x}^{\prime }\vert +{c}_{i},1\le i\le 2\right\}$ at infinity. In this paper, we show that two-end solutions are axially symmetric and monotonic if k 1 , k 2 satisfy k 1 − k 2 > 2 2 ${k}_{1}-{k}_{2}{ >}2\sqrt{2}$ . We also establish the nonexistence of two-end solution with k 1 , k 2 satisfying − 2 2 < k 2 < k 1 < 2 2 $-\frac{\sqrt{2}}{2}{< }{k}_{2}{< }{k}_{1}{< }\frac{\sqrt{2}}{2}$ or k 1 = − k 2 = 2 2 ${k}_{1}=-{k}_{2}=\frac{\sqrt{2}}{2}$ .

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