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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2509.19863 |
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| _version_ | 1866909804099796992 |
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| author | Li, Fengliu Vaira, Giusi Wei, Juncheng Wu, Yuanze |
| author_facet | Li, Fengliu Vaira, Giusi Wei, Juncheng Wu, Yuanze |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19863 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Brezis-Nirenberg problems: open questions and new results in dimension six Li, Fengliu Vaira, Giusi Wei, Juncheng Wu, Yuanze Analysis of PDEs 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. |
| title | On Brezis-Nirenberg problems: open questions and new results in dimension six |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.19863 |