Some remarks on the structure of finite Morse index solutions to the Allen-Cahn equation in $\mathbb{R}^2$

Abstract

For a solution of the Allen-Cahn equation in $\mathbb{R}^2$, under the natural linear growth energy bound, we show that the blowing down limit is unique. Furthermore, if the solution has finite Morse index, the blowing down limit satisfies the multiplicity one property.