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 ( − Δ ) s u = f ( u ) in R , where ( − Δ ) s stands for the s-Laplacian, s ∈ ( 0 , 1 ) . We introduce a suitable framework which allows, by means of regularity, to link the searching of such solutions into the existence of the ones of a semilinear problem in a suitable Hilbert space. Then by a bifurcation theory from eigenvalues of odd multiplicity and also variational method that avoid the constant solutions we get existence theorems which are lately enlightened with the analysis of some examples. In particular, multiplicity results for generalized Benjamin-Ono equation are obtained.