In this paper we establish existence and multiplicity solutions for a
class of Choquard-Kirchhoff type equations with variable exponents and
critical reaction. Because of the critical reaction, we can use the
concentration-compactness principle to deal with the lack of
compactness. The results are also based on the combination of the
mountain pass theorem and the Hardy-Littlewood-Sobolev inequality for
variable exponents. And we discuss the results in non-degenerate and
degenerate cases.