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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.20297 |
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| _version_ | 1866914498339667968 |
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| author | Guo, Yahong Zhang, Chilin |
| author_facet | Guo, Yahong Zhang, Chilin |
| contents | We focus on the classification of positive solutions to $(-Δ)^s u=\frac{x_n^α}{u^γ}$ in the half space with $γ>0$, subject to the Dirichlet condition. We show that when $-2s<α<(γ-1)s$, all positive solutions exhibit one-dimensional symmetry and are monotone increasing in $x_n$. Moreover, we provide a complete classification of all such one-dimensional solutions via their ``asymptotic $s$-order slope". When $α$ lies outside this range, we demonstrate the nonexistence of global positive solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20297 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification of solutions to a weighted singular fractional problem in the half space Guo, Yahong Zhang, Chilin Analysis of PDEs We focus on the classification of positive solutions to $(-Δ)^s u=\frac{x_n^α}{u^γ}$ in the half space with $γ>0$, subject to the Dirichlet condition. We show that when $-2s<α<(γ-1)s$, all positive solutions exhibit one-dimensional symmetry and are monotone increasing in $x_n$. Moreover, we provide a complete classification of all such one-dimensional solutions via their ``asymptotic $s$-order slope". When $α$ lies outside this range, we demonstrate the nonexistence of global positive solutions. |
| title | Classification of solutions to a weighted singular fractional problem in the half space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.20297 |