Blow-up of solutions of critical elliptic equations in three dimensions
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916299363319808 |
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| author | Frank, Rupert L. König, Tobias Kovařík, Hynek |
| author_facet | Frank, Rupert L. König, Tobias Kovařík, Hynek |
| contents | We describe the asymptotic behavior of positive solutions $u_ε$ of the equation $-Δu + au = 3\,u^{5-ε}$ in $Ω\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_ε$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation $-Δu + (a+εV) u = 3\,u^5$ in $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_10525 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Blow-up of solutions of critical elliptic equations in three dimensions Frank, Rupert L. König, Tobias Kovařík, Hynek Analysis of PDEs We describe the asymptotic behavior of positive solutions $u_ε$ of the equation $-Δu + au = 3\,u^{5-ε}$ in $Ω\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_ε$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation $-Δu + (a+εV) u = 3\,u^5$ in $Ω$. |
| title | Blow-up of solutions of critical elliptic equations in three dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2102.10525 |