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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.24100 |
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| _version_ | 1866917361117822976 |
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| author | Gao, Jinkai |
| author_facet | Gao, Jinkai |
| contents | In this paper, we consider the existence of positive solutions to the following slightly supercritical Choquard equation \begin{equation*}
\begin{cases}
-Δu=\displaystyle\Big(\int\limits_Ω\frac{u^{2^*_α+\varepsilon}(y)}{|x-y|^α}dy\Big)u^{2^*_α-1+\varepsilon},\quad u>0\ \ &\mbox{in}\ Ω,
\quad \ \ u=0 \ \ &\mbox{on}\ \partial Ω,
\end{cases} \end{equation*} where $N\geq 3$, $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$, $α\in (0,N)$, $2^*_α:=\frac{2N-α}{N-2}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and $\varepsilon>0$ is a small parameter. In contrast with the slightly subcritical Choquard equation studied by Chen and Wang (Calculus of Variations and Partial Differential Equations, 63:235, 2024), we find that there is no chance to construct a family of single-bubble solutions as $\varepsilon\to 0^{+}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonexistence of single-bubble solutions for a slightly supercritical Choquard equation Gao, Jinkai Analysis of PDEs In this paper, we consider the existence of positive solutions to the following slightly supercritical Choquard equation \begin{equation*} \begin{cases} -Δu=\displaystyle\Big(\int\limits_Ω\frac{u^{2^*_α+\varepsilon}(y)}{|x-y|^α}dy\Big)u^{2^*_α-1+\varepsilon},\quad u>0\ \ &\mbox{in}\ Ω, \quad \ \ u=0 \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $N\geq 3$, $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$, $α\in (0,N)$, $2^*_α:=\frac{2N-α}{N-2}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and $\varepsilon>0$ is a small parameter. In contrast with the slightly subcritical Choquard equation studied by Chen and Wang (Calculus of Variations and Partial Differential Equations, 63:235, 2024), we find that there is no chance to construct a family of single-bubble solutions as $\varepsilon\to 0^{+}$. |
| title | Nonexistence of single-bubble solutions for a slightly supercritical Choquard equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.24100 |