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

Abstract

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 to the equation which has growth rate $k$, i.e. $$\Vert u-H(\cdot -k\ln r+c_k){\Vert }_{L^{\infty }}\to 0$$ The main ingredients of our proof consist: (1) compactness of solutions with growth $k$, (2) moduli space theory of analytical variety of formal dimension one.

55 pages; comments welcome

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