On a fractional Schrödinger equation with periodic potential

Abstract In this paper, we first consider the spectrum of the fractional Schrodinger operator ( − Δ ) s + V ( x ) on R N , where s ∈ ( 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 ( − Δ ) 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 .

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