Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields

This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field: (aε2s + bε4s−3[u]2 A/ε ) (−∆)s A/εu + V (x)u = f(|u|2)u in R3, where ε > 0 is a small parameter, a, b > 0 are constants, s ∈ (4 3 , 1), (−∆)s A is the fractional magnetic Laplacian, A : R3 → R3 is a smooth magnetic potential, V : R3 → R is a positive continuous electric potential satisfying local conditions and f : R → R is a C1 subcritical nonlinearity. Applying penalization techniques, fractional Kato’s type inequality and Ljusternik-Schnirelmann theory, we relate the number of nontrivial solutions with the topology of the set where the potential V attains its minimum.