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

In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation $$-\mathrm{\Delta }u+\epsilon u=(I_{\alpha }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+g(u),\mathrm{i}\mathrm{n}{\mathbb{R}}^N,\text{eqno}(P_{\epsilon })$$ 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 $\epsilon \to \mathrm{\infty }$, 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 ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^2=a^2$. Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as $a\to 0$ and $a\to \mathrm{\infty }$.