Asymptotic profiles for Choquard equations with general critical nonlinearities

In this paper, we study asymptotic behavior of positive ground state solutions for the nonlinear Choquard equation: equation0. 1 - u+ u= (I_ F (u) ) F' (u), u H¹ (RN), equation where F (u) =|u|^N+{N-2}+G (u), N3 is an integer, I_ is the Riesz potential of order (0, N), and >0 is a parameter. Under some mild subcritical growth assumptions on G (u), we show that as, the ground state solutions of 0. 1, after a suitable rescaling, converge to a particular solution of the critical Choquard equation - u=N+N-2 (I_*|u|^N+{N-2}) |u|^N+{N-2-2}u. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the asymptotic behavior of G (u) at infinity and the space dimension N=3, N=4 or N5.