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arXiv · 1507.01830

A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane

Francois Hamel·Xavier Ros-Oton·Yannick Sire·Enrico Valdinoci
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Abstract

We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

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Francois Hamel, Xavier Ros-Oton, Yannick Sire, Enrico Valdinoci. 2015-07-07. A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. https://arxiv.org/abs/1507.01830

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