分数阶 Lane--Emden 猜想的证明

中文摘要 AI 辅助

设 $n\ge2$，$0<s<1$，且 $p,q>0$。我们证明分数阶 Lane--Emden 系统 \\[ (-\Delta)^s u=v^p,\qquad (-\Delta)^s v=u^q, \qquad u, v>0 \qquad\text{在 }\R^n\text{ 中} \\] 的 Liouville 定理。更精确地说，我们证明当 \\[ \frac1{p+1}+\frac1{q+1}>\frac{n-2s}{n} \\] 时，该系统不存在严格正的整体解。无需径向对称性、全局有界性、有限能量假设或无穷远处的指定衰减。我们的证明改编了最近为经典 Hénon--Lane--Emden 系统引入的局部化势核 virial 方法与源层区间压力方法。该方法在分数阶情形下同样有效的基本原因很简单：与 $(-\Delta)^s$ 相关联的 Riesz 核具有与 Newton 核相同的对数导数结构，只需将 $n-2$ 替换为 $n-2s$。然而，有两个真正的非局部点需要特殊处理。当 $2s\le1$ 时，单个核梯度项不是局部绝对可积的，因此在对角极限之前必须将两个方程合并。当 $2s>n-1$ 时，限制在一条直线上的核在无穷远处不可积，因此区间比较必须使用非负扩展积分来表述。在解决这些问题之后，局部化 virial 恒等式、一维区间对不等式和平均源层估计产生一个普遍的局部能量界。次临界缩放随后迫使同一能量发散，从而得到所需的矛盾。

英文摘要

Let $n\ge2$, $0<s<1$, and $p,q>0$. We prove a Liouville theorem for the fractional Lane--Emden system \[ (-Δ)^s u=v^p,\qquad (-Δ)^s v=u^q, \qquad u, v>0 \qquad\text{in }\R^n. \] More precisely, we show that the system does not have a strictly positive entire solution whenever \[ \frac1{p+1}+\frac1{q+1}>\frac{n-2s}{n}. \] No radial symmetry, global boundedness, finite-energy assumption, or prescribed decay at infinity is required. Our proof adapts the localized potential-kernel virial and source-layer interval-pressure method recently introduced for the classical Hénon--Lane--Emden system. The basic reason the method also works in the fractional setting is simple: the Riesz kernel associated with $(-Δ)^s$ has the same logarithmic derivative structure as the Newton kernel, with $n-2$ replaced by $n-2s$. However, two genuinely nonlocal points require special treatments. When $2s\le1$, the individual kernel-gradient terms are not locally absolutely integrable, so the two equations must be combined before the diagonal limit is taken. When $2s>n-1$, the kernel restricted to a line is not integrable at infinity, and the interval comparison must therefore be formulated using nonnegative extended integrals. After these issues are resolved, a localized virial identity, a one-dimensional interval-pair inequality, and an averaged source-layer estimate yield a universal local energy bound. The subcritical scaling then forces the same energy to diverge, giving the desired contradiction.

发表机构

• Zhejiang Normal University（浙江师范大学）
• The Chinese University of Hong Kong（香港中文大学）

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