(untitled) (Journal of Partial Differential Equations, 2025)
We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian (−∂_xx)^su(x)+∇F(u(x))=0 in R, where F : R^2→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.