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March 2011 Existence of infinitely many solutions for a class of Allen--Cahn equations in $\mathbb{R}^{2}$
Zheng Zhou
Osaka J. Math. 48(1): 51-67 (March 2011).

Abstract

In this paper we study the entire solutions of a class of periodic Allen--Cahn equations \begin{equation} -\Delta u(x,y)+a(x)W^{'}(u(x,y)) = 0{,}\quad (x,y) \in \mathbb{R}^{2}, \end{equation} where $a(x)\colon \mathbb{R} \to \mathbb{R}^{+}$ is a periodic, positive function and $W \in C^{2}(\mathbb{R}, \mathbb{R})$ is a double-well potential. We look for the entire solutions of the above equation with asymptotic conditions $u(x,y) \to \sigma_{\pm}$ as $x \to \pm\infty$ uniformly with respect to $y \in \mathbb{R}$. Via variational methods we find infinitely many solutions.

Citation

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Zheng Zhou. "Existence of infinitely many solutions for a class of Allen--Cahn equations in $\mathbb{R}^{2}$." Osaka J. Math. 48 (1) 51 - 67, March 2011.

Information

Published: March 2011
First available in Project Euclid: 22 March 2011

zbMATH: 1216.35057
MathSciNet: MR2802592

Subjects:
Primary: 35J60
Secondary: 35A15

Rights: Copyright © 2011 Osaka University and Osaka City University, Departments of Mathematics

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Vol.48 • No. 1 • March 2011
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