Asymptotic profiles for Choquard equations with combined attractive nonlinearities

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation<br /> (PE) -Au+Eu = (Ia* |u)|u|P(2)u+u4-2u, in R-N,> N+a NN where N >= 3 is an integer, p <euro> [+, Na], q <euro> (2, 2), I, is the Riesz potential of order a <euro> (0, N) and epsilon > 0 is a parameter. We show that as epsilon -> 0 (resp. epsilon -> infinity), the ground state solutions of (P), after appropriate rescalings dependent on parameter regimes, converge in H1 (RN) to particular solutions of five different limit equations. We also establish a sharp asymptotic characterisation of such rescalings, and the precise asymptotic behaviour of us (0), ||Vue ||2, || ||2, SRN (I&* |ue|P)|ue|P and || ||2, which depend in a non-trivial way on the exponents p, q and the space dimension N. Further, we discuss a connection of our results with a mass constrained problem, associated to (P) with normalization constraint fp |u|2 = c2. As a consequence of the main results, we obtain the existence, multiplicity and precise asymptotic behaviour of positive normalized solutions of such a problem as c -> 0 and c -> infinity. (c) 2024 The Author (s). Published by Elsevier Inc. This is an open access article under the CC BY license(http://creativecommons.org/licenses/by/4.0/).