Multiplicity and concentration results for fractional Kirchhoff equation with magnetic field

In the paper, we considered the fractional Kirchhoff equation with magnetic field (a epsilon(2s) + b epsilon(4s-3)[u](A/epsilon)(2) )(- Delta)(A/e)(s) u + V(x)u = f (vertical bar u vertical bar(2))u in R-3, where epsilon is a small parameter, a, b are positive constants, s is an element of (3/4, 1), A : R-3 -> R-3 is a Holder continuous magnetic potential with exponent alpha is an element of (0, 1], [u](A/epsilon)(2) = integral integral(6)(R) vertical bar u(x)-epsilon"(x-y)A/epsilon(x+y/2)u(y)(2)vertical bar/vertical bar x-y vertical bar 3+2s dxdy. 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.