On variational characterization of four-end solutions of the Allen-Cahn equation in the plane

详细信息    查看全文

• 作者：Changfeng Gui^a ; ^b ; ^{changfeng.gui@utsa.edu" class="auth_mail" title="E-mail the corresponding author} ; Yong Liu^c ; ^{liuyong@ncepu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Juncheng Wei^d ; ^{jcwei@math.ubc.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B32 ; 35C07 ; 35J20 ; 35R09 ; 35R11 ; 45G05 ; 47G10
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：271
• 期：10
• 页码：2673-2700
• 全文大小：436 K

文摘

In this paper a novel variational method is developed to construct four-end solutions in 35e71e0e2d" title="Click to view the MathML source">R²${\mathbb{R}}^2$ for the Allen–Cahn equation. Four-end solutions have been constructed by Del Pino, Kowalczyk, Pacard and Wei when the angle θ   of the ends is close to π/2$\pi /2$ or 0 using the Lyapunov–Schmid reduction method, and later by Kowalczyk, Liu and Pacard using a continuation method for general θ∈(0,π/2)$\theta \in (0,\pi /2)$. By a special mountain pass argument in a restricted space, namely a set of monotone paths of monotone functions, a family of solutions in bounded domain is constructed with very good control of their nodal sets, which then leads to a four-end solution after sending the domains to 35e71e0e2d" title="Click to view the MathML source">R²${\mathbb{R}}^2$. In this way, not only a family of four end solutions is constructed for any angle θ∈(0,π/2)$\theta \in (0,\pi /2)$. The Morse index of such a four-end solution is also shown to be one.