Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation

This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$in$$\mathbb{R}$^{2m}$,where$L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone ${(x’, x’‘) \in $\mathbb{R}$^m \times $\mathbb{R}$^m , : , |x’| = |x’'|}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in “narrow” sets. The existence of the solution was already proved in part I of this work.