The role of the boundary in the existence of blow-up solutions for a doubly critical elliptic problem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911941460492288 |
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| author | Cruz-Blázquez, Sergio |
| author_facet | Cruz-Blázquez, Sergio |
| contents | In this paper we consider a doubly critical nonlinear elliptic problem with Neumann boundary conditions. The existence of blow-up solutions for this problem is related to the blow-up analysis of the classical geometric problem of prescribing negative scalar curvature $K=-1$ on a domain of $\R^n$ and mean curvature $H=D(n(n-1))^{-1/2}$, for some constant $D>1$, on its boundary, via a conformal change of the metric. Assuming that $n\geq6$ and $D>\sqrt{(n+1)/(n-1)}$, we establish the existence of a positive solution which concentrates around an elliptic boundary point which is a nondegenerate critical point of the original mean curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08024 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The role of the boundary in the existence of blow-up solutions for a doubly critical elliptic problem Cruz-Blázquez, Sergio Analysis of PDEs 35B33, 35B44, 35J60 In this paper we consider a doubly critical nonlinear elliptic problem with Neumann boundary conditions. The existence of blow-up solutions for this problem is related to the blow-up analysis of the classical geometric problem of prescribing negative scalar curvature $K=-1$ on a domain of $\R^n$ and mean curvature $H=D(n(n-1))^{-1/2}$, for some constant $D>1$, on its boundary, via a conformal change of the metric. Assuming that $n\geq6$ and $D>\sqrt{(n+1)/(n-1)}$, we establish the existence of a positive solution which concentrates around an elliptic boundary point which is a nondegenerate critical point of the original mean curvature. |
| title | The role of the boundary in the existence of blow-up solutions for a doubly critical elliptic problem |
| topic | Analysis of PDEs 35B33, 35B44, 35J60 |
| url | https://arxiv.org/abs/2312.08024 |