On the solvability for a p-k-Hessian inequality

TL;DR

We study the inequality $F_{k,p}[u]\ge (-u)^{\alpha}$ in $\mathbb{R}^n$, where $F_{k,p}$ is the $p$-$k$-Hessian operator. The authors develop non-symmetric $\sigma_k$-based identities to relate $F_{k,p}[u]$ to a transformed Hessian and employ a local integral–iteration framework, including a constructive negative subsolution for the supercritical regime. For $pk<n$ they establish a sharp threshold $k_{p,*}=\frac{n(p-1)k}{n-pk}$ such that no negative solution exists in $\Phi^{p,k}(\mathbb{R}^n)$ when $\alpha\le k_{p,*}$, while $\alpha>k_{p,*}$ admits negative solutions. This sharp nonexistence/existence dichotomy extends classical results for the Laplacian, $p$-Laplacian, and $k$-Hessian to the general $p$-$k$-Hessian setting and provides a versatile method for fully nonlinear inequalities in the whole space.

Abstract

In this paper, we discuss the solvability of a p-k-Hessian entire inequality. We prove that the inequality with sub-lower-critical exponent admits no negative solutions. Moreover, the exponent is sharp. The proof is based on choosing suitable test functions and integrating by parts.

Theorems & Definitions (13)

• Theorem 1.1: Serrin-Zou SerrinZou2002Acta
• Theorem 1.2: Phuc-Verbitsky PhucVerbitsky2006CPDEPhucVerbitsky2008Ann, Ou Ouqianzhong2010MAA
• Theorem 1.3
• Proposition 2.1: Pietra, Gavitone and Xia PietraGavitoneXia2021Adv
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof