A critical fractional Choquard-Kirchhoff problem with magnetic field

In this paper, we are interested in a fractional Choquard-Kirchhoff-type problem involving an external magnetic potential and a critical nonlinearity M(parallel to u parallel to(2)(s,A))[(-Delta)(A)(s) u + u] = lambda integral(RN) F(vertical bar u vertical bar(2))/vertical bar x-y vertical bar(alpha) dy f(vertical bar u vertical bar(2))u + vertical bar u vertical bar(2s*-2u) in R-N, parallel to u parallel to s,A = (integral integral(R2N) vertical bar u(x) - e(i(x-y).A(x+y/2))u(y)vertical bar(2)/vertical bar x-y vertical bar(N+2s) dxdy + integral(RN) vertical bar u vertical bar(2)dx)1/2 , where N > 2s with 0 < s < 1, M is the Kirchhoff function, A is the magnetic potential, (-Delta)(A)(s) is the fractional magnetic operator, f is a continuous function, F(vertical bar u vertical bar) = integral(vertical bar u vertical bar)(0) f(t)dt, lambda > 0 is a parameter, 0 < alpha < min{N,4s} and 2(s)* = 2N/N - 2s is the critical exponent of fractional Sobolev space. We first establish a fractional version of the concentration-compactness principle with magnetic field. Then, together with the mountain pass theorem, we obtain the existence of nontrivial radial solutions for the above problem in non-degenerate and degenerate cases.