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

Abstract

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 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.