Singular semilinear elliptic equations in half-spaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917765714018304 |
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| author | Le, Phuong |
| author_facet | Le, Phuong |
| contents | We prove the monotonicity of positive solutions to the problem $-Δu = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some situations, we can derive a precise estimate on the blow-up rate of $\frac{\partial u}{\partialη}$ as $x_N \to 0^+$, where $(η,e_N)>0$, and obtain a classification result. The main tools we use are the method of moving planes and the sliding method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00365 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular semilinear elliptic equations in half-spaces Le, Phuong Analysis of PDEs 35J61, 35J75, 35B06, 35B09 We prove the monotonicity of positive solutions to the problem $-Δu = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some situations, we can derive a precise estimate on the blow-up rate of $\frac{\partial u}{\partialη}$ as $x_N \to 0^+$, where $(η,e_N)>0$, and obtain a classification result. The main tools we use are the method of moving planes and the sliding method. |
| title | Singular semilinear elliptic equations in half-spaces |
| topic | Analysis of PDEs 35J61, 35J75, 35B06, 35B09 |
| url | https://arxiv.org/abs/2409.00365 |