Layer and stable solutions to a nonlocal model

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