Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones

For all n 1, we are interested in bounded solutions of the Allen–Cahn equation u+u−u3 = 0 which are defined in all Rn+1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n + 1 8, we prove the existence of stable solutions of the Allen–Cahn equation whose zero sets are not hyperplanes. © 2012 Published by Elsevier Inc.