Asymptotic profiles for Choquard equations with general critical nonlinearities

TL;DR

This paper investigates the asymptotic behavior of positive ground state solutions for a class of nonlinear Choquard equations with critical nonlinearities, revealing how solutions behave as a parameter tends to infinity and establishing sharp asymptotic characterizations.

Contribution

The paper provides a novel analysis of the asymptotic profiles of ground state solutions for Choquard equations with critical nonlinearities, including a detailed characterization depending on the growth of additional nonlinear terms.

Findings

Solutions converge to a critical Choquard solution after rescaling as epsilon tends to infinity.

The asymptotic profile depends on the growth of G(u) at infinity and the space dimension.

A sharp asymptotic characterization of the rescaling is established.

Abstract

In this paper, we study asymptotic behavior of positive ground state solutions for the nonlinear Choquard equation: \begin{equation}\label{0.1} -\Delta u+\varepsilon u=\big(I_{\alpha}\ast F(u)\big)F'(u),\quad u\in H^1(\mathbb R^N), \end{equation} where $F(u)=|u|^{\frac{N+\alpha}{N-2}}+G(u)$, $N\geq3$ is an integer, $I_{\alpha}$ is the Riesz potential of order $\alpha\in(0,N)$, and $\varepsilon>0$ is a parameter. Under some mild subcritical growth assumptions on $G(u)$, we show that as $\varepsilon \to \infty$, the ground state solutions of \eqref{0.1}, after a suitable rescaling, converge to a particular solution of the critical Choquard equation $-\Delta u=\frac{N+\alpha}{N-2}(I_{\alpha}*|u|^{\frac{N+\alpha}{N-2}})|u|^{\frac{N+\alpha}{N-2}-2}u$. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the asymptotic behavior of $G(u)$ at infinity and the space dimension $N=3$, $N=4$ or $N\geq5$.