Asymptotic profiles for Choquard equations with combined attractive nonlinearities

TL;DR

This paper investigates the asymptotic behavior of ground state solutions to a nonlinear Choquard equation with combined nonlinearities, revealing how solutions behave as a parameter approaches zero or infinity, and connecting these results to normalized solutions.

Contribution

It provides a detailed asymptotic analysis of ground state solutions for the Choquard equation with combined nonlinearities, including convergence, sharp characterizations, and behavior of key quantities.

Findings

Solutions converge to limit equations as parameters tend to zero or infinity.

Sharp asymptotic characterizations of solutions and their norms.

Existence and multiplicity results for normalized solutions as mass varies.

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 $c\to \infty$.