On a fractional Schrodinger equation with periodic potential

In this paper, we first consider the spectrum of the fractional Schrodinger operator (-Delta)(s)+ V(x) on R-N, where s is an element of (0, 1) and V(x) be continuous, periodic in x. Using a new nonlocal normal derivative, we prove that the operator has purely continuous spectrum which is bounded below and consists of closed disjoint intervals. Then when 0 belongs to a spectral gap of (-Delta)(s)+ V(x), we establish an existence result for the fractional Schrodinger equation via a new linking theorem. It is worth to mention that the nonlinear term in our problem does not satisfy the periodicity and the existence result is even new for the case s = 1. (C) 2019 Published by Elsevier Ltd.