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

TL;DR

This paper studies saddle-shaped solutions to a class of semilinear integro-differential equations related to the Allen-Cahn model, proving uniqueness, asymptotic behavior, and symmetry properties in a high-dimensional setting.

Contribution

It establishes the uniqueness, asymptotic behavior, and symmetry of saddle-shaped solutions to nonlocal Allen-Cahn equations, extending previous existence results.

Findings

Proved uniqueness of saddle-shaped solutions.

Analyzed asymptotic behavior at infinity.

Established one-dimensional symmetry of solutions.

Abstract

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.