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

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.

arXiv commentThe paper supercedes the papers arXiv:1505.06919 and arXiv:1506.00109. Final version, accepted for publication in Ann. Inst. H. Poincaré Anal. Non Linéaire

References in corpus (3)

• Harnack type inequality for positive solution of some integral equation
• A one-dimensional symmetry result for solutions of an integral equation of convolution type
• Entire solutions to semilinear nonlocal equations in $\RR^2$