Layer and stable solutions to a nonlocal model

Abstract

We study the layer and stable solutions of nonlocal problem $ \begin{equation*} -\Delta u+F'(u)\left( -\Delta \right) ^{s}F(u)+G'(u) = 0\text{ in }\mathbb{R}^{n} \end{equation*} $ where $ F\in C_{{\text{loc}}}^2( \mathbb R) $ satisfies $ F(0) = 0 $ and $ G $ is a double well potential. For $ n = 2,s>0 $ and $ n = 3, $ $ s\geq 1/2, $ we establish the 1-d symmetry of layer solutions for this equation. When $ n = 2 $ and $ F' $ is bounded away from zero, we prove the 1-d symmetry of stable solutions for this equation. Using a different approach, we also prove the 1-d symmetry of stable solutions for$ \begin{equation*} F'(u)\left( -\Delta \right) ^{s}F(u)+G'(u) = 0\text{ in }\mathbb{R}^{2}. \end{equation*} $