Asymptotic profiles of ground state solutions for Choquard equations with a general local perturbation

2024/05/05 by Shiwang Ma, Vitaly Moroz, Ma, Shiwang +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2405.02877

openalex publication_date 2024/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation -Δu+ε u=(I_α∗ |u|^p)|u|^p-2u+ g(u), \rm in \mathbb R^N, \eqno(P_ε) where N≥ 3 is an integer, p=(N+α)/(N), or (N+α)/(N-2), I_α is the Riesz potential and ε>0 is a parameter. Under some mild conditions on g(u), we show that as ε→ ∞, after \em a suitable rescaling the ground state solutions of (P_ε) converge to a particular solution of some limit equations, and establish a sharp asymptotic characterisation of such a rescaling, which depend in a non-trivial way on the asymptotic behavior of the function g(s) at infinity and the space dimension N. Based on this study, we also present some results on the existence and asymptotic behaviors of positive normalized solutions of (P_ε) with the normalization constraint ∫_{\mathbb R^N}|u|²=a². Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as a→ 0 and a→ ∞.

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