Critical Exponent Elliptic Equations on the Half-Space: Uniqueness and Explicit Solutions
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909762597158912 |
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| author | Nouri, Azam |
| author_facet | Nouri, Azam |
| contents | We prove that all positive solutions of $-Δu = u^{\frac{2n}{n-2}}$ on the upper half space $\mathbb{R}^n_{+}$ (for $n \geq 3$) satisfying the boundary condition $D_{x_n}u = -u^{\frac{n}{n-2}}$ are of the form $u(x) = a \left( \fracλ{λ^2 + |x-y|^2} \right)^{\frac{n-2}{2}}$, where $a = a(n)$, $λ> 0$, and $y = (y_1, \ldots, y_n)$ is a point in the lower half-space with $y_n < 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical Exponent Elliptic Equations on the Half-Space: Uniqueness and Explicit Solutions Nouri, Azam Analysis of PDEs We prove that all positive solutions of $-Δu = u^{\frac{2n}{n-2}}$ on the upper half space $\mathbb{R}^n_{+}$ (for $n \geq 3$) satisfying the boundary condition $D_{x_n}u = -u^{\frac{n}{n-2}}$ are of the form $u(x) = a \left( \fracλ{λ^2 + |x-y|^2} \right)^{\frac{n-2}{2}}$, where $a = a(n)$, $λ> 0$, and $y = (y_1, \ldots, y_n)$ is a point in the lower half-space with $y_n < 0$. |
| title | Critical Exponent Elliptic Equations on the Half-Space: Uniqueness and Explicit Solutions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.12218 |