Optimal Rigidity and Classification Results for the $k$-Hessian Equation of Lane--Emden Type

In this paper, we establish optimal Liouville theorems and classification results for the $k$-Hessian Lane--Emden equation $$\sigma_k(-D^2u)=u^p\quad\text{in }\R^n,\qquad -D^2u\in\overline{\Gamma_k},\qquad u\geq 0,$$ 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 completely closed this gap and proved 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 proved that nontrivial nonnegative $C^2$ entire solution must be the Aubin-Talenti type bubble without any assumptions for $2k<n\leq 4k$, under boundedness assumption for $n=4k+1$ and $4k+2$, and under some appropriate integral growth conditions or pointwise asymptotic behavior assumptions for any $n>4k$ and the limiting case $n=2k$. In particular, we provide a fully nonlinear counterparts of the classical Liouville and classification theorems of Gidas--Spruck, Gidas--Ni--Nirenberg, and Caffarelli--Gidas--Spruck.