2023/02/27 by Ma, Shiwang, Moroz, Vitaly
#35B25 #35B40 #35J60 #35J91 #35Q55 #35R09 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2302.13727
We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation -Δu+ε u=(Iα∗ |u|p)|u|p-2u+ |u|q-2u \rm in \mathbb RN, where N≥ 3 is an integer, p∈ [(N+α)/(N), (N+α)/(N-2)], q∈ (2,(2N)/(N-2)], Iα is the Riesz potential and ε>0 is a parameter. We show that as ε→ 0 (resp. ε→ ∞), after a suitable rescaling the ground state solutions of (Pε) converge in H1(\mathbb RN) to a particular solution of some limit equations. We also establish a sharp asymptotic characterisation of such a rescaling, and the exact asymptotic behaviours of uε(0), ‖∇ uε‖22, ‖uε‖22, ∫\mathbb 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. We also discuss a connection of our results with an associated mass constrained problem with normalization constraint ∫\mathbb RN|u|2=c2. As a consequence of the main results, we obtain the existence, multiplicity and exact asymptotic behaviour of positive normalized solutions of such a problem as c→ 0 and c→ ∞.