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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.06613 |
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| _version_ | 1866912634708688896 |
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| author | Li, Kui Li, Mingxiang Wei, Juncheng |
| author_facet | Li, Kui Li, Mingxiang Wei, Juncheng |
| contents | We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system
\[
-Δu = v^p, \quad -Δv = u^q, \quad x \in \mathbb{R}^n\]
in the subcritical regime. By employing an Obata-type integral inequality, Picone's identity, and exploiting the scaling invariance of the system, we prove that the conjecture holds for any dimension $n \geq 5$ and exponents satisfying $p\geq 1,q\geq 1$, and
\[
\frac{1}{p+1} + \frac{1}{q+1} \geq 1 - \frac{2}{n} + \frac{4}{n^2}.
\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a new region for the Lane-Emden conjecture in higher dimensions Li, Kui Li, Mingxiang Wei, Juncheng Analysis of PDEs 35J60 We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system \[ -Δu = v^p, \quad -Δv = u^q, \quad x \in \mathbb{R}^n\] in the subcritical regime. By employing an Obata-type integral inequality, Picone's identity, and exploiting the scaling invariance of the system, we prove that the conjecture holds for any dimension $n \geq 5$ and exponents satisfying $p\geq 1,q\geq 1$, and \[ \frac{1}{p+1} + \frac{1}{q+1} \geq 1 - \frac{2}{n} + \frac{4}{n^2}. \] |
| title | On a new region for the Lane-Emden conjecture in higher dimensions |
| topic | Analysis of PDEs 35J60 |
| url | https://arxiv.org/abs/2510.06613 |