Multiplicity and concentration results for fractional Kirchhoff equation with magnetic field

In the paper, we considered the fractional Kirchhoff equation with magnetic field (aε2s+bε4s−3[u]A/ε2)(−Δ)A/εsu+V(x)u=f(|u|2)uin R3, where ε is a small parameter, a, b are positive constants, s∈(34,1), A:R3→R3 is a Hölder continuous magnetic potential with exponent α∈(0,1], [u]A/ε2=∬R6|u(x)−εı(x−y)⋅Aε(x+y2)u(y)|2|x−y|3+2sdxdy. With proper assumptions on V and f, we develop a new approach to prove the Palais–Smale condition for the corresponding energy functional and establish the multiplicity and concentration of solutions by using perturbation techniques and Ljusternik–Schnirelmann theory.