发表机构
University of Chinese Academy of Sciences（中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了 Lane-Emden 系统在圆盘形指数区域内无正经典解（除非临界点），通过构造向量场和积分估计，填补了现有准则的空白。
AI 中文摘要
我们获得了 Lane-Emden 猜想的新结果，证明了 Lane-Emden 系统 $$\begin{cases} -\Delta u=v^p,& x\in \mathbb{R}^N,\\\\ -\Delta v=u^q,& x\in \mathbb{R}^N,\end{cases}$$ 的 Liouville 定理，其中 $N\ge3$，$p,q>0$，$pq>1$，且 \\[ \left(p-\frac2{N-2}\right)^2+ \left(q-\frac2{N-2}\right)^2\le\frac{2N^2}{(N-2)^2}, \\] 当 $(p,q)$ 位于该圆盘内时，系统不存在正经典解，除非 $p=q=(N+2)/(N-2)$，且不施加任何增长或衰减假设，该圆盘与临界双曲线相切。特别地，对于 $N\ge5$，该圆盘包含一个非空开集，该开集未被 Busca-Manásevich~\cite{BM} 和 Souplet~\cite{Souplet} 的准则覆盖，也未包含在 Li-Li-Wei~\cite{LLW} 的显式条件中。在通过已知 Liouville 准则缩减指数范围后，我们首先构造向量场以推导散度不等式，然后应用 Sobolev 嵌入定理、Gagliardo-Nirenberg 插值不等式和 Young 不等式，给出对任意正解 $(u,v)$ 的均匀局部积分界 $\int_{B_1}(u^rv^{2p}+u^{2q}v^s)dx\le C(N,p,q)$（适用于适当的权重），最后通过缩放得出正经典解的不存在性。
英文摘要
We obtain new results on Lane-Emden conjecture to prove a Liouville theorem for the Lane-Emden system $$\begin{cases} -Δu=v^p,& x\in \mathbb{R}^N,\\ -Δv=u^q,& x\in \mathbb{R}^N,\end{cases}$$ with $N\ge3$, $p,q>0$, $pq>1$, and \[ \left(p-\frac2{N-2}\right)^2+ \left(q-\frac2{N-2}\right)^2\le\frac{2N^2}{(N-2)^2}, \] 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 $N\ge5$, 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 $\int_{B_1}(u^rv^{2p}+u^{2q}v^s)dx\le C(N,p,q)$ for suitable weights for any positive solution $(u,v)$, finally the nonexistence of positive classical solutions follows by scaling.
