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(∥u∥s,A2)[(−Δ) Asu + u] = λ∫ℝN F(|u|2) |x − y|αdyf(|u|2)u + |u|2s∗−2uin ℝN, ∥u∥s,A = ∬ℝ2N|u(x) − ei(x−y)⋅A(x+y 2 )u(y)|2 |x − y|N+2s dxdy + ∫ℝN|u|2dx1/2, where N > 2s with 0 0 is a parameter, 0 < α 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.