Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two
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2025
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| author | Gao, Jinkai Li, Xinfu Ma, Shiwang |
| author_facet | Gao, Jinkai Li, Xinfu Ma, Shiwang |
| contents | In this paper, we are interested in the following planar Choquard equation \begin{equation*}
\begin{cases}
-Δu=\displaystyle\left(\int\limits_Ω\frac{u^{p+1}(y)}{|x-y|^α}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ Ω,
\quad \ \ u=0, \ \ &\mbox{on}\ \partial Ω,
\end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^2$, $α\in (0,2)$ and $p>1$ is a positive parameter. Unlike the higher-dimensional case, we prove that the least energy solutions $u_{p}$ neither blow up nor vanish, and develop only one peak as $p\to+\infty$ under suitable assumptions on $Ω$. In contrast, the modified solutions $pu_p$ exhibit blow-up behavior analogous to that observed in higher dimensions. Furthermore, as $α\to 0$, the main results of this paper become consistent with the known conclusions for the corresponding Lane-Emden equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02139 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two Gao, Jinkai Li, Xinfu Ma, Shiwang Analysis of PDEs In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -Δu=\displaystyle\left(\int\limits_Ω\frac{u^{p+1}(y)}{|x-y|^α}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ Ω, \quad \ \ u=0, \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^2$, $α\in (0,2)$ and $p>1$ is a positive parameter. Unlike the higher-dimensional case, we prove that the least energy solutions $u_{p}$ neither blow up nor vanish, and develop only one peak as $p\to+\infty$ under suitable assumptions on $Ω$. In contrast, the modified solutions $pu_p$ exhibit blow-up behavior analogous to that observed in higher dimensions. Furthermore, as $α\to 0$, the main results of this paper become consistent with the known conclusions for the corresponding Lane-Emden equation. |
| title | Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.02139 |