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

1 Universitat Politècnica de Catalunya and BGSMath, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain
2 School of Mathematics, The University of Edinburgh, Peter Tait Guthrie Road, EH9 3FD Edinburgh, UK

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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.
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