Asymptotic profiles for Choquard equations with general critical nonlinearities

2024/05/12 by Liu, Xiaonan, Ma, Shiwang, Wang, Yachen
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.07149

Abstract

In this paper, we study asymptotic behavior of positive ground state solutions for the nonlinear Choquard equation: -Δu+ε u=(I_α∗ F(u))F'(u), u∈ H¹(\mathbb R^N), where F(u)=|u|^(N+α)/(N-2)+G(u), N≥3 is an integer, I_α 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 \eqref0.1, after a suitable rescaling, converge to a particular solution of the critical Choquard equation -Δu=(N+α)/(N-2)(I_α*|u|^(N+α)/(N-2))|u|^{(N+α)/(N-2)-2}u. 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.

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