Lawson cones and the Allen-Cahn equation

In this paper we discuss nondegeneracy and stability properties of some special minimal hypersurfaces which are asymptotic to a given Lawson cone $C_{m,n}$, for $m,n\ge 2$. Then we use such hypersurfaces to construct solutions to the Allen-Cahn equation $-\mathrm{\Delta }u=u-u^3$ in ${\mathbb{R}}^{N+1}$, $N+1\ge 8$, whose zero level set has exactly $k\ge 2$ connected components and with infinite Morse index.