Periodic solutions of a semilinear elliptic equation with a fractional Laplacian

Abstract

We consider periodic solutions of the following problem associated with the fractional Laplacian $$(-\partial _{xx})^s u(x) + F'(u(x))=0,\quad u(x)=u(x+T),\quad \text{ in } \, \mathbb {R}, $$ where \((-\partial _{xx})^s\) denotes the usual fractional Laplace operator with \(0<s<1\). The primitive function F of the nonlinear term is a smooth double-well potential. We prove the existence of periodic solutions with large period T using variational methods. An estimate of the energy of the periodic solutions is also established.