Existence thresholds and limit profiles of ground states for lower critical Choquard equations with general nonlinearities

TL;DR

This paper investigates the existence, thresholds, and asymptotic behavior of ground states for a class of nonlinear Choquard equations with general nonlinearities, revealing critical parameters and limit profiles as the frequency parameter approaches zero.

Contribution

It provides a sharp threshold criterion for ground state existence and characterizes the asymptotic limit profiles of solutions under various nonlinear growth conditions.

Findings

Existence thresholds depend on the nonlinearity exponent q.

Ground states converge to solutions of a critical equation as ε→0.

A novel asymptotic characterization of rescaled ground states is established.

Abstract

In this paper, we study the existence, non-existence and asymptotic behavior of positive ground states 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 $F(u)=|u|^{\frac{N+\alpha}{N}}+G(u)$ with $G(u)=\int_0^ug(s)ds$, $N\geq3$ is an integer, $I_{\alpha}$ is the Riesz potential of order $\alpha\in(0,N)$ and $\varepsilon>0$ is a frequency parameter. Under some mild subcritical growth assumptions on $g\in C([0,\infty), [0,\infty))$, we establish a sharp threshold result for the existence of ground states, and an asymptotic characterization of the ground state solutions as $\varepsilon\to 0$. In particular, if $g(s)\sim s^{q-1}$ as $s\to 0$ for some $q\in (\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2})$, then if $q<\frac{N+\alpha+4}{N}$, \eqref{0.1} admits a ground state for all $\varepsilon>0$, and if $q\ge \frac{N+\alpha+4}{N}$, then a threshold phenomena occur: there exists $\varepsilon_q>0$ such that \eqref{0.1} has no ground state for $\varepsilon\in (0,\varepsilon_q)$ and admits a ground state for $\varepsilon>\varepsilon_q$. If $g(s)\simeq as^{q-1}$ as $s\to 0$ for some $a>0$ and $q\in (\frac{N+\alpha}{N}, \min\{\frac{N+\alpha}{N-2}, \frac{N+\alpha+4}{N}\})$, we show that as $\varepsilon \to 0$, the ground state solutions of \eqref{0.1}, after a suitable rescaling, converges in $H^1(\mathbb R^N)$ to a particular solution of the Hardy-Littlewood-Sobolev critical equation $u=\frac{N+\alpha}{N}(I_{\alpha}*|u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}-2}u$. It turns out that the limit profiles are determined solely by the locations of $(a,q)$ in $(0,+\infty)\times (\frac{N+\alpha}{N}, \min\{\frac{N+\alpha}{N-2}, \frac{N+\alpha+4}{N}\})$. We also establish a novel sharp asymptotic characterization of such a rescaling.