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

Abstract

In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation $$ -Δu+\varepsilon u=(I_α\ast |u|^{p})|u|^{p-2}u+ g(u), \quad {\rm in} \ \mathbb R^N, \eqno(P_\varepsilon) $$ where $N\ge 3$ is an integer, $p=\frac{N+\alpha }{N}$, or $\frac{N+\alpha }{N-2}$, $I_{\alpha }$ is the Riesz potential and $\epsilon >0$ is a parameter. Under some mild conditions on $g(u)$, we show that as $\epsilon \to \infty$, after {\em a suitable rescaling} the ground state solutions of $(P_{\epsilon })$ 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 $(P_{\epsilon })$ with the normalization constraint ${\int }_{{\mathbb{R}}^N}|u{|}^2=a^2$. Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as $a\to 0$ and $a\to \infty$.

41 pages, 0 figure. arXiv admin note: substantial text overlap with arXiv:2302.13727