Asymptotic profiles for Choquard equations with combined attractive nonlinearities

Abstract

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation $$ -\Delta u+\varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u \quad {\rm in} \ \mathbb R^N, $$ where $N\ge 3$ is an integer, $p\in [\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}]$, $q\in (2,\frac{2N}{N-2}]$, $I_\alpha$ is the Riesz potential and $\varepsilon>0$ is a parameter. We show that as $\varepsilon\to 0$ (resp. $\varepsilon\to \infty$), after a suitable rescaling the ground state solutions of $(P_\varepsilon)$ converge in $H^1(\mathbb R^N)$ to a particular solution of some limit equations. We also establish a sharp asymptotic characterisation of such a rescaling, and the exact asymptotic behaviours of $u_\varepsilon(0), \|\nabla u_\varepsilon\|_2^2, \|u_\varepsilon\|_2^2, \int_{\mathbb R^N}(I_\alpha\ast |u_\varepsilon|^p)|u_\varepsilon|^p$ and $\|u_\varepsilon\|_q^q$, which depend in a non-trivial way on the exponents $p, q$ and the space dimension $N$. We also discuss a connection of our results with an associated mass constrained problem with normalization constraint $\int_{\mathbb R^N}|u|^2=c^2$. As a consequence of the main results, we obtain the existence, multiplicity and exact asymptotic behaviour of positive normalized solutions of such a problem as $c\to 0$ and $c\to \infty$.