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Classification of positive solutions to the critical Lane-Emden system
Abstract
We classify all positive classical solutions of the Lane--Emden system −Δu=vp in ℝN, −Δv=uq in ℝN, where N≥3 and p,q>0 satisfy 1/(p+1)+1/(q+1)=(N−2)/N.Without assuming boundedness, decay at infinity, or finite energy, we prove that every such solution has the form u(x)=λN/(q+1)U(λ(x−a)), v(x)=λN/(p+1)V(λ(x−a)), for some a∈ ℝN and λ>0, where (U,V) is the positive radial ground state centered at the origin and normalized by U(0)=1. Consequently, both components are radially symmetric about the same point and satisfy ∫ℝNuq+1 dx = ∫ℝNvp+1 dx<∞. Our result extends the classical classification of positive solutions of the critical Sobolev equation by Caffarelli, Gidas and Spruck and Chen and Li to the critical Lane-Emden system.
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BibTeX
@misc{guo2026classification,
title = {{Classification of positive solutions to the critical Lane-Emden system}},
author = {Yuxia Guo and Yichen Hu and Shaolong Peng and Tingfeng Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27875},
eprint = {2610.27875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27875},
}
RIS
TY - UNPB
TI - Classification of positive solutions to the critical Lane-Emden system
AU - Yuxia Guo
AU - Yichen Hu
AU - Shaolong Peng
AU - Tingfeng Yuan
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27875
UR - https://arxiv.org/abs/2610.27875
AB - We classify all positive classical solutions of the Lane--Emden system −Δu=v^p in ℝ^N, −Δv=u^q in ℝ^N, where N≥3 and p,q>0 satisfy 1/(p+1)+1/(q+1)=(N−2)/N.Without assuming boundedness, decay at infinity, or finite energy, we prove that every such solution has the form u(x)=λ^N/(q+1)U(λ(x−a)), v(x)=λ^N/(p+1)V(λ(x−a)), for some a∈ ℝ^N and λ>0, where (U,V) is the positive radial ground state centered at the origin and normalized by U(0)=1. Consequently, both components are radially symmetric about the same point and satisfy ∫_ℝ^Nu^q+1 dx = ∫_ℝ^Nv^p+1 dx<∞. Our result extends the classical classification of positive solutions of the critical Sobolev equation by Caffarelli, Gidas and Spruck and Chen and Li to the critical Lane-Emden system.
ER -