Existence of infinitely many stationary layered solutions in R^2 for a class of periodic Allen Cahn Equations.

We consider a class of periodic Allen-Cahn equations \begin{equation}\tag{$1$} -\Delta u(x,y)+a(x,y)W'(u(x,y))=0,\quad (x,y)\in\R^{2} \end{equation} where $a:\R^2\to\R^2$ is an even, periodic, positive function and $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-b^{2})^{2}$. We show, via variational methods, that there exist infinitely many solutions, distinct up to periodic translations, of $(1)$ asymptotic as $x\to\pm\infty$ to the pure states $\pm b$, i.e., solutions satisfying the boundary conditions \begin{equation}\label{eq:et}\tag{$2$} \lim_{x\to\pm\infty}u(x,y)=\pm b, \quad \hbox{uniformly in $y \in \R.$} \end{equation} In fact, we prove the existence of solutions of $(1)$-$(2)$ which are periodic in the $y$ variable and if such solutions are finite modulo periodic translations, we can prove the existence of infinitely many (modulo periodic translations) solutions of $(1)$-$(2)$ asymptotic to different periodic solutions as $y \to \pm\infty$.