Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in R^{3}

We consider a class of semilinear elliptic equations of the form \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\R^{3}, \end{equation} where $a:\R\to\R$ is a periodic, positive, even function and, in the simplest case, $W:\R\to\R$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the associated one dimensional heteroclinic problem we show, via variational methods the existence of infinitely many geometrically distinct solutions $u$ of (\ref{eq:abs}) verifying $u(x,y,z)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $(y,z)\in\R^{2}$ and such that $\partial_{y}u\not\equiv0$, $\partial_{z}u\not\equiv0$ in $\R^{3}$.