article

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

Authors: Francesca Alessio

Issue: Volume 62 (2013), Issue 5, 1535-1564

Abstract:

$We consider a class of a semilinear elliptic system of the form \[ (0.1)\qquad-\Delta u(x,y)+\nabla W(u(x,y))=0,\quad(x,y)\in\mathbb{R}^2, \] where $W:\mathbb{R}^2\to\mathbb{R}$ is a double well nonnegative 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 (0.1) has infinitely many layered solutions.$