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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.19863 |
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Table of Contents:
- In this paper, we consider the Brezis-Nirenberg problem \begin{equation*} \left\{\begin{aligned} &-Δu = λu+|u|^{2^*-2}u, \quad &\mbox{in}\,Ω,\\ &u=0,\quad &\mbox{on}\, \partialΩ, \end{aligned}\right. \end{equation*} where $Ω$ is a smoothly bounded domain of $\mathbb R^N$ with $N\geq 3$, $λ>0$ is a parameter and $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent. We first recall the history of the Brezis-Nirenberg problem and then provide new results of it in dimension six. Finally, we also list some open questions on the Brezis-Nirenberg problem.