  This paper is devoted to study a fractional Choquard problem with slightly
subcritical exponents on bounded domains. When the exponent of the convolution
type nonlinearity tends to the fractional critical one in the sense of
Hardy-Littlewood-Sobolev inequality, we obtain the existence of multiple
positive solutions via Lusternik-Schnirelmann category and nonlocal global
compactness. Moreover, we prove that the topology of the domain furnishes a
lower bound for the number of positive solutions.
