Abstract

In this paper we study the existence of multiple-layer solutions to the elliptic Allen-Cahn equation in hyperbolic space: -Delta(n)(H) u + F'(u) = 0; here F is a nonnegative double-well potential with nondegenerate minima. We prove that for any collection of widely separated, nonintersecting hyperplanes in H-n, there is a solution to this equation which has a nodal set very close to this collection of hyperplanes. Unlike the corresponding problem in R-n, there are no constraints beyond the separation parameter.