Existence thresholds and limit profiles of ground states for lower critical Choquard equations with general nonlinearities
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917303081238528 |
|---|---|
| author | Ma, Shiwang Wang, Yachen |
| author_facet | Ma, Shiwang Wang, Yachen |
| contents | 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 $F(u)=|u|^{\frac{N+α}{N}}+G(u)$ with $G(u)=\int_0^ug(s)ds$, $N\geq3$ is an integer, $I_α$ is the Riesz potential of order $α\in(0,N)$ 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 $q\in (\frac{N+α}{N}, \frac{N+α}{N-2})$, then if $q<\frac{N+α+4}{N}$, \eqref{0.1} admits a ground state for all $\varepsilon>0$, and if $q\ge \frac{N+α+4}{N}$, 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 $q\in (\frac{N+α}{N}, \min\{\frac{N+α}{N-2}, \frac{N+α+4}{N}\})$, 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 $u=\frac{N+α}{N}(I_α*|u|^{\frac{N+α}{N}})|u|^{\frac{N+α}{N}-2}u$. It turns out that the limit profiles are determined solely by the locations of $(a,q)$ in $(0,+\infty)\times (\frac{N+α}{N}, \min\{\frac{N+α}{N-2}, \frac{N+α+4}{N}\})$. We also establish a novel sharp asymptotic characterization of such a rescaling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01078 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence thresholds and limit profiles of ground states for lower critical Choquard equations with general nonlinearities Ma, Shiwang Wang, Yachen Analysis of PDEs 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 $F(u)=|u|^{\frac{N+α}{N}}+G(u)$ with $G(u)=\int_0^ug(s)ds$, $N\geq3$ is an integer, $I_α$ is the Riesz potential of order $α\in(0,N)$ 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 $q\in (\frac{N+α}{N}, \frac{N+α}{N-2})$, then if $q<\frac{N+α+4}{N}$, \eqref{0.1} admits a ground state for all $\varepsilon>0$, and if $q\ge \frac{N+α+4}{N}$, 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 $q\in (\frac{N+α}{N}, \min\{\frac{N+α}{N-2}, \frac{N+α+4}{N}\})$, 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 $u=\frac{N+α}{N}(I_α*|u|^{\frac{N+α}{N}})|u|^{\frac{N+α}{N}-2}u$. It turns out that the limit profiles are determined solely by the locations of $(a,q)$ in $(0,+\infty)\times (\frac{N+α}{N}, \min\{\frac{N+α}{N-2}, \frac{N+α+4}{N}\})$. We also establish a novel sharp asymptotic characterization of such a rescaling. |
| title | Existence thresholds and limit profiles of ground states for lower critical Choquard equations with general nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.01078 |