Fractional magnetic Schrodinger-Kirchhoff problems with convolution and critical nonlinearities

In this paper, we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrodinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: {epsilon M-2s([u](s,A)(2))(-Delta)(A)(s)u + V(x)u = (vertical bar x vertical bar(-alpha) * F(vertical bar u vertical bar(2))) f (vertical bar u vertical bar(2))u + vertical bar u vertical bar(2s)*(-2) u, x is an element of R-N, u(x) -> 0, as vertical bar x vertical bar -> infinity, where (-Delta)s A is the fractional magnetic operator with 0 < s < 1, 2(s)* = 2N/(N - 2s), alpha < min{N, 4s}, M : R-0(+) -> R-0(+) is a continuous function, A : R-N -> R-N is the magnetic potential, F(vertical bar u vertical bar) =integral(vertical bar u vertical bar)(0) f(t)dt, and epsilon > 0 is a positive parameter. The electric potential V is an element of C(R-N, R-0(+)) satisfies V(x) = 0 in some region of R-N, which means that this is the critical frequency case. We first prove the (PS)(c) condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term M can vanish at zero.