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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.09250 |
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| _version_ | 1866913732671569920 |
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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 $$ -Δu=λu+|u|^{\frac{4}{N-2}}u,\quad\mbox{in}\,\, Ω,\quad u=0,\quad\mbox{on}\,\, \partialΩ, $$ where $λ\in\mathbb{R}$, $Ω\subset\mathbb R^N$ is a bounded domain with smooth boundary $\partialΩ$ and $N\geq3$. We prove that every eigenvalue of the Laplacian operator $-Δ$ with the Dirichlet boundary is a concentration value of the Brezis-Nirenberg problem in dimensions $N=4$ and $N=5$ by constructing bubbling solutions with precisely asymptotic profiles via the Ljapunov-Schmidt reduction arguments. Our results suggest that the bubbling phenomenon of the Brezis-Nirenberg problem in dimensions $N=4$ and $N=5$ as the parameter $λ$ is close to the eigenvalues are governed by crucial functions related to the eigenfunctions, which has not been observed yet in the literature to our best knowledge. Moreover, as the parameter $λ$ is close to the eigenvalues, there are arbitrary number of multi-bump bubbing solutions in dimension $N=4$ while, there are only finitely many number of multi-bump bubbing solutions in dimension $N=5$, which are also new findings to our best knowledge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09250 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of bubbling solutions of the Brezis-Nirenberg problem in general bounded domains (I): the dimensions 4 and 5 Li, Fengliu Vaira, Giusi Wei, Juncheng Wu, Yuanze Analysis of PDEs In this paper, we consider the Brezis-Nirenberg problem $$ -Δu=λu+|u|^{\frac{4}{N-2}}u,\quad\mbox{in}\,\, Ω,\quad u=0,\quad\mbox{on}\,\, \partialΩ, $$ where $λ\in\mathbb{R}$, $Ω\subset\mathbb R^N$ is a bounded domain with smooth boundary $\partialΩ$ and $N\geq3$. We prove that every eigenvalue of the Laplacian operator $-Δ$ with the Dirichlet boundary is a concentration value of the Brezis-Nirenberg problem in dimensions $N=4$ and $N=5$ by constructing bubbling solutions with precisely asymptotic profiles via the Ljapunov-Schmidt reduction arguments. Our results suggest that the bubbling phenomenon of the Brezis-Nirenberg problem in dimensions $N=4$ and $N=5$ as the parameter $λ$ is close to the eigenvalues are governed by crucial functions related to the eigenfunctions, which has not been observed yet in the literature to our best knowledge. Moreover, as the parameter $λ$ is close to the eigenvalues, there are arbitrary number of multi-bump bubbing solutions in dimension $N=4$ while, there are only finitely many number of multi-bump bubbing solutions in dimension $N=5$, which are also new findings to our best knowledge. |
| title | Construction of bubbling solutions of the Brezis-Nirenberg problem in general bounded domains (I): the dimensions 4 and 5 |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.09250 |