Classification of Anti-symmetric Solutions to Fractional Lane-Emden System in Borderline Case
Abstract
We investigate the existence and non-existence of anti-symmetric solutions to the fractional Lane--Emden system in the half-space ℝn+: (−Δ)s u = vp & in ℝn+, (−Δ)s v = uq & in ℝn+, where 0<s<1/2,p,q>0,2s+p=1, with u,v nonnegative in ℝn+ and anti-symmetric with respect to ∂ℝn+. Li and Zhuo [2023, Sci. China Math.] previously proved non-existence in L2s, as well as non-existence in the extended space L2s+1 when 1−2s<p,q<(n+2s)/(n−2s) and existence in L2s+1 when p+2s<1,q+2s<1. In this paper, we fill the remaining borderline case p=1−2s in L2s+1 space. We establish a non-existence result for 1−2s=p≤ q≤(n+2s)/(n−2s) and an existence result for 0<q<p=1−2s. For the non-existence result, we combine a maximum principle for anti-symmetric functions with the method of moving planes to derive a Liouville--type theorem when 1−2s<q≤(n+2s)/(n−2s). We then iterate the lower-bound estimates to obtain a contradiction through blow-up. To prove the existence of nontrivial solutions when 0<q<1−2s, we construct a nonhomogeneous solution by a monotone iteration scheme in L2s+1 as well as analyze a homogeneous profile. Surprisingly, the homogeneous profile serves as a universal lower bound for all solutions.
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BibTeX
@misc{li2026classification,
title = {{Classification of Anti-symmetric Solutions to Fractional Lane-Emden System in Borderline Case}},
author = {Congming Li and Zixuan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39165},
eprint = {2610.39165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39165},
}
RIS
TY - UNPB
TI - Classification of Anti-symmetric Solutions to Fractional Lane-Emden System in Borderline Case
AU - Congming Li
AU - Zixuan Wang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39165
UR - https://arxiv.org/abs/2610.39165
AB - We investigate the existence and non-existence of anti-symmetric solutions to the fractional Lane--Emden system in the half-space ℝ^n_+: (−Δ)^s u = v^p & in ℝ^n_+, (−Δ)^s v = u^q & in ℝ^n_+, where 0<s<1/2,p,q>0,2s+p=1, with u,v nonnegative in ℝ^n_+ and anti-symmetric with respect to ∂ℝ^n_+. Li and Zhuo [2023, Sci. China Math.] previously proved non-existence in L_2s, as well as non-existence in the extended space L_2s+1 when 1−2s<p,q<(n+2s)/(n−2s) and existence in L_2s+1 when p+2s<1,q+2s<1. In this paper, we fill the remaining borderline case p=1−2s in L_2s+1 space. We establish a non-existence result for 1−2s=p≤ q≤(n+2s)/(n−2s) and an existence result for 0<q<p=1−2s. For the non-existence result, we combine a maximum principle for anti-symmetric functions with the method of moving planes to derive a Liouville--type theorem when 1−2s<q≤(n+2s)/(n−2s). We then iterate the lower-bound estimates to obtain a contradiction through blow-up. To prove the existence of nontrivial solutions when 0<q<1−2s, we construct a nonhomogeneous solution by a monotone iteration scheme in L_2s+1 as well as analyze a homogeneous profile. Surprisingly, the homogeneous profile serves as a universal lower bound for all solutions.
ER -