Periodic solutions for the one-dimensional fractional Laplacian

Abstract

In this paper we are concerned with the construction of periodic solutions of the nonlocal problem $(-\Delta)^s u= f(u)$ in $\mathbb{R}$, where $(-\Delta)^s$ stands for the $s$-Laplacian, $s\in (0,1)$. We introduce a suitable framework which allows to reduce the search for such solutions to the resolution of a boundary value problem in a suitable Hilbert space, thereby making it possible to reach for the usual tools of nonlinear analysis, like bifurcation theory or variational methods. We obtain some existence theorems which are lately enlightened with the analysis of some examples.