New result on Lane-Emden conjecture with exponents in a disk

Abstract

We obtain new results on Lane-Emden conjecture to prove a Liouville theorem for the Lane-Emden system −Δu=v^p,& x∈ ℝ^N, −Δv=u^q,& x∈ ℝ^N, with N≥3, p,q>0, pq>1, and (p−2/(N−2))²+ (q−2/(N−2))²≤2N²/((N−2)²), the system has no positive classical solution as (p,q) in the disk unless p=q=(N+2)/(N−2), no growth or decay assumptions are imposed, and the disk is tangent to the critical hyperbola. Especially, for N≥5, the disk contains a nonempty open set not covered by the criteria of Busca-Manásevich~\cite{BM} and Souplet~\cite{Souplet}, or by the explicit conditions of Li-Li-Wei~\cite{LLW}. After reducing the exponent range by known Liouville criteria, we first construct vector fields to derive a divergence inequality, then apply Sobolev embedding theorem, Gagliardo-Nirenberg interpolation inequality, and Young's inequality to give a uniform local integral bound ∫_B₁(u^rv^2p+u^2qv^s)dx≤ C(N,p,q) for suitable weights for any positive solution (u,v), finally the nonexistence of positive classical solutions follows by scaling.

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BibTeX RIS

BibTeX

@misc{lu2026result,
  title = {{New result on Lane-Emden conjecture with exponents in a disk}},
  author = {Haoyang Lu and Zhitao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09526},
  eprint = {2610.09526},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09526},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - New result on Lane-Emden conjecture with exponents in a disk
AU  - Haoyang Lu
AU  - Zhitao Zhang
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09526
UR  - https://arxiv.org/abs/2610.09526
AB  - We obtain new results on Lane-Emden conjecture to prove a Liouville theorem for the Lane-Emden system −Δu=v^p,& x∈ ℝ^N, −Δv=u^q,& x∈ ℝ^N, with N≥3, p,q>0, pq>1, and (p−2/(N−2))^2+ (q−2/(N−2))^2≤2N^2/((N−2)^2), the system has no positive classical solution as (p,q) in the disk unless p=q=(N+2)/(N−2), no growth or decay assumptions are imposed, and the disk is tangent to the critical hyperbola. Especially, for N≥5, the disk contains a nonempty open set not covered by the criteria of Busca-Manásevich~\cite{BM} and Souplet~\cite{Souplet}, or by the explicit conditions of Li-Li-Wei~\cite{LLW}. After reducing the exponent range by known Liouville criteria, we first construct vector fields to derive a divergence inequality, then apply Sobolev embedding theorem, Gagliardo-Nirenberg interpolation inequality, and Young's inequality to give a uniform local integral bound ∫_B_1(u^rv^2p+u^2qv^s)dx≤ C(N,p,q) for suitable weights for any positive solution (u,v), finally the nonexistence of positive classical solutions follows by scaling.
ER  -

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