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