Stationary layered solutions for a system of Allen-Cahn type equations

Abstract

We consider a class of semilinear elliptic system of the form $-\Delta u(x,y)+\nabla W(u(x,y))=0,(x,y)\in {\mathbb{R}}^2$ where $W:{\mathbb{R}}^2\to \mathbb{R}$ is a double well non negative symmetric potential. We show, via variational methods, that if the set of solutions to the one dimensional system $-\ddot{q}(x)+\nabla W(q(x))=0,x\in \mathbb{R}$, which connect the two minima of $W$ as $x\to \pm \infty$ has a discrete structure, then the given system has infinitely many layered solutions.

References in corpus (2)

• On an elliptic system with symmetric potential possessing two global minima
• Even Symmetry of Some Entire Solutions to the Allen-Cahn Equation in Two Dimensions

Cited by in corpus (1)

• An energy constrained method for the existence of layered type solutions of NLS equations