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

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 $\{(x^{\prime },x^{\prime \prime })\in {\mathbb{R}}^m\times {\mathbb{R}}^m:|x^{\prime }|=|x^{\prime \prime }|\}$, 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.

arXiv commentTo appear in Mathematics in Engineering

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Cited by in corpus (1)

• Semilinear integro-differential equations, I: odd solutions with respect to the Simons cone