Choquard equations with critical nonlinearities

In this paper, we study the Brezis–Nirenberg type problem for Choquard equations in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] are the critical exponents in the sense of Hardy–Littlewood–Sobolev inequality and [Formula: see text] is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Pohožaev constraint method, we obtain a positive and radially nonincreasing ground-state solution in [Formula: see text] for the problem. To the end, the regularity and the Pohožaev identity of solutions to a general Choquard equation are obtained.