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| Natura: | Preprint |
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2022
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| Accesso online: | https://arxiv.org/abs/2211.00595 |
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| _version_ | 1866915686935166976 |
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| author | König, Tobias Laurain, Paul |
| author_facet | König, Tobias Laurain, Paul |
| contents | For a bounded set $Ω\subset \mathbb R^N$ and a perturbation $V \in C^1(\overlineΩ)$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -Δu_ε+ εV = N(N-2) u_ε^\frac{N+2}{N-2} \] for dimensions $N \geq 4$, which are non-critical in the sense of the Brezis--Nirenberg problem.
For the general case of multiple concentration points, we prove that concentration points are isolated and characterize the vector of these points as a critical point of a suitable function derived from the Green's function of $-Δ$ on $Ω$. Moreover, we give the leading order expression of the concentration speed. This paper, with a recent one by the authors (arXiv:2208.12337) in dimension $N = 3$, gives a complete picture of blow-up phenomena in the Brezis-Nirenberg framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00595 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem König, Tobias Laurain, Paul Analysis of PDEs For a bounded set $Ω\subset \mathbb R^N$ and a perturbation $V \in C^1(\overlineΩ)$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -Δu_ε+ εV = N(N-2) u_ε^\frac{N+2}{N-2} \] for dimensions $N \geq 4$, which are non-critical in the sense of the Brezis--Nirenberg problem. For the general case of multiple concentration points, we prove that concentration points are isolated and characterize the vector of these points as a critical point of a suitable function derived from the Green's function of $-Δ$ on $Ω$. Moreover, we give the leading order expression of the concentration speed. This paper, with a recent one by the authors (arXiv:2208.12337) in dimension $N = 3$, gives a complete picture of blow-up phenomena in the Brezis-Nirenberg framework. |
| title | Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2211.00595 |