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 -Δ u+ε u=(I_α |u|ᵖ)|u|ᵖ⁻²u+ g(u), in \ RN, (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 { 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 ∫RN|u|²=a². Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as a→ 0 and a→ ∞.