Asymptotic profiles for Choquard equations with combined attractive nonlinearities: the locally critical case

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We study asymptotic behavior of positive ground state solutions of the nonlinear Choquard equation with a Sobolev critical attractive local perturbation where N≥3 is an integer, p∈(N+αN,N+αN-2), 2∗=2NN-2 is the Sobolev critical exponent, Iα is the Riesz potential of order α∈(0,N) and ε>00$$\\end{document}]]> is a parameter. We show that as ε→∞, after suitable rescalings the ground state solutions uε of (Pε) converge to a particular solution of the critical local Emden–Fowler equation. The rescalings are implicit and depend in a non-trivial way on the exponent p and the space dimension N=3,4 or N≥5. We establish a sharp asymptotic characterisation of such rescalings, as well as the blow-up rates or asymptotics of the L2 and other relevant norm of uε. As a follow up of our main results, we also obtain the existence, multiplicity and asymptotic behaviour of positive normalized solutions of a mass constrained problem associated to (Pε) with mass normalization constraint ∫RN|u|2=c2, as c→0 and c→∞.