Asymptotic profiles for Choquard equations with combined attractive nonlinearities
Shiwang Ma, Vitaly Moroz
Abstract
We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation(Pε)−Δu+εu=(Iα⁎|u|p)|u|p−2u+|u|q−2u,inRN,where N≥3 is an integer, p∈[N+αN,N+αN−2], q∈(2,2NN−2), Iα is the Riesz potential of order α∈(0,N) and ε>0 is a parameter. We show that as ε→0 (resp. ε→∞), the ground state solutions of (Pε), 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 uε(0), ‖∇uε‖22, ‖uε‖22, ∫RN(Iα⁎|uε|p)|uε|p and ‖uε‖qq, 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 (Pε) with normalization constraint ∫RN|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→∞.