Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity

Abstract In this paper we focus our attention on the following nonlinear fractional Schrodinger equation with magnetic field e 2 s ( − Δ ) A / e s u + V ( x ) u = f ( | u | 2 ) u  in  R N , where e > 0 is a parameter, s ∈ ( 0 , 1 ) , N ≥ 3 , ( − Δ ) A s is the fractional magnetic Laplacian, V : R N → R and A : R N → R N are continuous potentials and f : R N → R is a subcritical nonlinearity. By applying variational methods and Ljusternick–Schnirelmann theory, we prove existence and multiplicity of solutions for e small.