Liouville-type Theorems for Stable Solutions of the Hénon-Lane-Emden System
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911325885562880 |
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| author | Huang, Long-Han Zou, Wenming |
| author_facet | Huang, Long-Han Zou, Wenming |
| contents | We investigate the Hénon-Lane-Emden system defined by $- Δu=|x|^a |v|^{p-1}v$ and $- Δv=|x|^b |u|^{q-1}u$ in $\mathbb{R}^N \!\setminus\! \{0\}$. We begin by establishing a general Liouville-type theorem for the subcritical case. Then we prove that the Hénon-Lane-Emden conjecture is valid for solutions stable outside a compact set, provided that $0 < \min\,\{p, q\} < 1$, or $0 \leq a - b \leq (N-2)(p - q)$, or $N \leq \frac{2(p+q+2)}{pq-1} + 10$. Additional Liouville-type theorems for the subcritical case are also obtained. Furthermore, we address the supercritical case. To our knowledge, these results constitute the first Liouville-type theorems for this class of solutions in the Hénon-Lane-Emden system. As a by-product, several existing results in the literature are refined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville-type Theorems for Stable Solutions of the Hénon-Lane-Emden System Huang, Long-Han Zou, Wenming Analysis of PDEs 35B09, 35B40, 35B33 We investigate the Hénon-Lane-Emden system defined by $- Δu=|x|^a |v|^{p-1}v$ and $- Δv=|x|^b |u|^{q-1}u$ in $\mathbb{R}^N \!\setminus\! \{0\}$. We begin by establishing a general Liouville-type theorem for the subcritical case. Then we prove that the Hénon-Lane-Emden conjecture is valid for solutions stable outside a compact set, provided that $0 < \min\,\{p, q\} < 1$, or $0 \leq a - b \leq (N-2)(p - q)$, or $N \leq \frac{2(p+q+2)}{pq-1} + 10$. Additional Liouville-type theorems for the subcritical case are also obtained. Furthermore, we address the supercritical case. To our knowledge, these results constitute the first Liouville-type theorems for this class of solutions in the Hénon-Lane-Emden system. As a by-product, several existing results in the literature are refined. |
| title | Liouville-type Theorems for Stable Solutions of the Hénon-Lane-Emden System |
| topic | Analysis of PDEs 35B09, 35B40, 35B33 |
| url | https://arxiv.org/abs/2512.16566 |