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: {equation}{0.1} -Δ u+ε u=(Iα F(u))F'(u), u∈ H^1( R^N), {equation} where F(u)=|u|N+α/N-2+G(u), N≥3 is an integer, Iα is the Riesz potential of order α∈(0,N), and ε>0 is a parameter. Under some mild subcritical growth assumptions on $G(u)$, we show that as ε → ∞, the ground state solutions of {0.1}, after a suitable rescaling, converge to a particular solution of the critical Choquard equation -Δ u=N+α/N-2(Iα*|u|N+α/N-2)|u|N+α/N-2-2u. 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 N≥5.