Multiplicity of layered solutions for Allen-Cahn systems with symmetric double well potential

Abstract

We study the existence of solutions $u:\R^{3}\to\R^{2}$ for the semilinear elliptic systems \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+\nabla W(u(x,y,z))=0, \end{equation} where $W:\R^{2}\to\R$ is a double well symmetric potential. We use variational methods to show, under generic non degenerate properties of the set of one dimensional heteroclinic connections between the two minima $\a_{\pm}$ of $W$, that (\ref{eq:abs}) has infinitely many geometrically distinct solutions $u\in C^{2}(\R^{3},\R^{2})$ which satisfy $u(x,y,z)\to \a_{\pm}$ as ${x\to\pm\infty}$ uniformly with respect to $(y,z)\in\R^{2}$ and which exhibit dihedral symmetries with respect to the variables $y$ and $z$. We also characterize the asymptotic behaviour of these solutions as $|(y,z)|\to +\infty$.