Stationary layered solutions in R^2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations \begin{equation} -\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}, \end{equation} where $W\in\CC^{2}(\R,\R)$ is a multiple-well potential and $a\in\CC(\R,\R)$ is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as $x\to\pm\infty$ corresponding to minima of $W$. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation $-\ddot q(x)+a(x)W'(q(x))=0,$ $x\in\R,$ the equation (0.1) has solutions which depends on both the variables $x$ and $y$. In contrast, when $a$ is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on $x$.