Multiple Periodic Solutions of Allen-Cahn System Involving Fractional Laplacian

Abstract

We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian $$(−∂_{xx})^su(x)+∇F(u(x))=0 \quad {\rm in} \quad \mathbb{R},$$ where $F$ : $\mathbb{R}^2$→$\mathbb{R}$ is a smooth double-well potential. For the case that $F$ is even in its two variables we obtain the existence of more and more periodic solutions with large period, by using Clark’s theorem. For the case that $F$ is only even in its the second variable and the origin is a saddle critical point of $F$, we give two periodic solutions by using Morse theory.