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

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} -Δu+\varepsilon u=\big(I_α\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 $G(u)={\int }_0^{u}g(s)ds$, $N\ge 3$ is an integer, $I_{\alpha }$ is the Riesz potential of order $\alpha \in (0,N)$ and $\epsilon >0$ 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 $\epsilon \to 0$. 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 $\epsilon >0$, and if $q\ge \frac{N+\alpha +4}{N}$, then a threshold phenomena occur: there exists ${\epsilon }_q>0$ such that \eqref{0.1} has no ground state for $\epsilon \in (0,{\epsilon }_q)$ and admits a ground state for $\epsilon >{\epsilon }_q$. If $g(s)\simeq a{s}^{q-1}$ 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 $\epsilon \to 0$, 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 $u=\frac{N+\alpha }{N}(I_{\alpha }\ast |u{|}^{\frac{N+\alpha }{N}})|u{|}^{\frac{N+\alpha }{N}-2}u$. 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.

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