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

Abstract

We study the existence of solutions $u:{\mathbb{R}}^3\to {\mathbb{R}}^2$ for the semilinear elliptic systems \begin{equation}\label{eq:abs} -Δu(x,y,z)+\nabla W(u(x,y,z))=0, \end{equation} where $W:{\mathbb{R}}^2\to \mathbb{R}$ 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 $u\in C^2({\mathbb{R}}^3,{\mathbb{R}}^2)$ which satisfy $u(x,y,z)\to \a_{\pm}$ as $x\to \pm \infty$ uniformly with respect to $(y,z)\in {\mathbb{R}}^2$ and which exhibit dihedral symmetries with respect to the variables $y$ and $z$. We also characterize the asymptotic behaviour of these solutions as $|(y,z)|\to +\infty$.

References in corpus (1)

• On an elliptic system with symmetric potential possessing two global minima