发表机构
Universidad Nacional de Colombia, Sede Medellín; Università degli studi di Bari Aldo Moro（哥伦比亚国立大学麦德林校区; 巴里阿尔多·莫罗大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在一般非退化假设下构造了高维Allen-Cahn方程的解，其零水平集渐近于任意给定极小锥，推广了Pacard-Wei的稳定解结果，并计算了Morse指标。
AI 中文摘要
我们在$\mathbb{R}^{N+1}$（其中$N\ge 3$）中构造了Allen-Cahn方程$\Delta u+u-u^3=0$的解，其零水平集在无穷远处渐近于给定的极小锥。我们的构造相当一般，因为我们仅使用了适当的非退化假设；既不要求底层锥的稳定性，也不要求其对称性。此外，我们计算了此类解的Morse指标。特别地，我们的主要结果推广了Pacard和Wei（2013）关于高维Allen-Cahn方程稳定解的结果。更确切地说，我们对Pacard和Wei论文中提出的一个评论给出了肯定回答，该评论涉及放宽底层锥为面积最小化或至少稳定的假设的可能性。作为推论，我们还获得了具有无限Morse指标且零水平集连通的解。
英文摘要
We construct solutions to the Allen-Cahn equation $Δu+u-u^3=0$ in $\mathbb{R}^{N+1}$, with $N\ge 3$, whose zero level set is asymptotic (at infinity) to a given minimal cone. Our construction is quite general, since we only use a suitable nondegeneracy assumption; neither stability nor symmetry of the underlying cone is required. Moreover, we calculate the morse index of such solutions. In particular, our main result generalizes a result by Pacard and Wei (2013) dealing with stable solutions to the Allen-Cahn equation in high dimensions. More precisely, we give a positive answer to a remark raised in the paper by Pacard and Wei about the possibility to relax the assumption for the underlying cone is area-minimizing or at least stable. As a corollary, we also obtain solutions with infinite morse index and connected zero level set.
