Existence of infinitely many solutions for a class of Allen--Cahn equations in $\mathbb{R}^{2}$

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.