AI 中文总结
针对Lane--Emden型k-Hessian方程，填补了p∈(0,k]到k<p≤p_-间的研究空白，证明了p_-<p<p*时的最优Liouville定理，确定p*为尖锐阈值，还给出临界情形p=p*的分类结果，得到经典定理的完全非线性对应。
AI 中文摘要
本文针对 Lane--Emden 型 $k$-Hessian 方程 $\sigma_k(-D^2u)=u^p$（在 $\mathbb{R}^n$ 内，满足 $-D^2u\in\overline{\Gamma_k}$、$u\geq0$，其中 $2\leq k<\frac{n}{2}$ 且 $p>0$）建立了最优 Liouville 定理与分类结果。令 $p_- = \frac{nk}{n-2k}$，临界 Hessian--Sobolev 指数 $p_* = \frac{(n+2)k}{n-2k}$，Phuc 和 Verbitsky 证明了当 $k<p\leq p_-$ 时正解不存在，Ou 随后覆盖了 $p\in(0,k]$ 的情形。我们完全填补了这一空白，对任意 $p_-<p<p_*$ 证明了最优 Liouville 定理：任意非负 $C^2$ 整体解必恒为零；同时对非负局部有界的 Hessian 测度弱解也证明了最优 Liouville 定理。这确定了临界指数 $p_*$ 为尖锐的 Liouville 阈值，因为当 $p\geq p_*$ 时存在径向正解。对于临界情形 $p=p_*$，我们证明：当 $2k<n\leq 4k$ 时，非平凡非负 $C^2$ 整体解必为 Aubin-Talenti 型 bubble；当 $n=4k+1$ 和 $4k+2$ 时，在有界性假设下成立；当 $n>4k$ 的任意情形及极限情形 $n=2k$ 时，在适当的积分增长条件或逐点渐近行为假设下成立。特别地，我们给出了 Gidas--Spruck、Gidas--Ni--Nirenberg 及 Caffarelli--Gidas--Spruck 经典 Liouville 与分类定理的完全非线性对应结果。
英文摘要
In this paper, we establish optimal Liouville theorems and classification results for the \(k\)-Hessian Lane--Emden equation \[ σ_k(-D^2u)=u^p\quad\text{in }\R^n,\qquad -D^2u\in\overline{Γ_k},\qquad u\geq 0, \] where \(2\leq k<\frac{n}{2}\) and $p>0$. Let $p_- = \frac{nk}{n-2k}$ and the critical Hessian--Sobolev exponent $p_* = \frac{(n+2)k}{n-2k}$. Phuc and Verbitsky proved nonexistence of positive solutions for \(k<p\leq p_-\), while Ou subsequently covered the cases \(p\in(0,k]\). We close this gap and prove the optimal Liouville theorem for any \(p_-<p<p_*\): any nonnegative \(C^2\) entire solution must be identically zero. We also prove the optimal Liouville theorem for nonnegative locally bounded Hessian-measure weak solutions. This identifies the critical exponent \(p_*\) as the sharp Liouville threshold, since radial positive solutions exist for \(p\geq p_*\). For the critical case \(p=p_*\), we prove that every nontrivial nonnegative \(C^2\) entire solution is a Hessian--Sobolev bubble for every \(n>2k\), without any additional assumption. For the limiting case \(n=2k\), we classify finite-mass solutions to the $\frac{n}{2}$-Hessian Liouville equation under a proper asymptotic condition $u(x)\rightarrow-\infty$ as $|x|\rightarrow\infty$. In particular, we provide the fully nonlinear counterparts of the classical Liouville and classification theorems of Gidas--Spruck, Gidas--Ni--Nirenberg, and Caffarelli--Gidas--Spruck.
CommentsThis version updates the previously uploaded version of Aug. 5, 2026
