--- title: P-k-Hessian Equations in Nonlinear Analysis url: https://www.emergentmind.com/topics/p-k-hessian-type-equations type: topic --- # P-k-Hessian Equations in Nonlinear Analysis P–k–Hessian type equations are a class of highly nonlinear partial differential equations (PDEs) defined via the composition of gradient-dependent and Hessian-based nonlinearities, generalizing both the p-Laplacian and classical k-Hessian operators. They are central in both real and complex geometric analysis, nonlinear potential theory, and nonlinear elliptic and parabolic PDE, and are studied in the context of the Euclidean space, Riemannian and Hermitian manifolds, as well as in pluripotential and variational frameworks. ## 1. Definition of the $p$–$k$–Hessian Operator The real $p$–$k$–Hessian operator, denoted $F_{k,p}[u]$, is defined for a smooth function $u:\R^n\to\R$ as follows. Set \[ X(x) = |Du|^{p-2} Du \in \R^n, \quad p > 1, \] and let $DX$ denote the Jacobian matrix of $X$: \[ DX = D(|Du|^{p-2} Du). \] Let $\lambda(DX) = (\lambda_1, ..., \lambda_n)$ be the eigenvalues of $DX$, and define the $k$-th elementary symmetric function: \[ \sigma_k(\lambda) = \sum_{1 \le i_1 < \cdots < i_k \le n} \lambda_{i_1} \cdots \lambda_{i_k}. \] Then \[ F_{k,p}[u](x) = \sigma_k(D(|Du|^{p-2} Du)(x)). \] Special cases include: - $k=1$: $F_{1,p}[u]$ is the $p$–Laplacian $\Delta_p u = \operatorname{div}(|Du|^{p-2} Du)$. - $p=2$: $F_{k,2}[u]=\sigma_k(D^2u)$ is the classical $k$-Hessian operator [2503.00895]. A function $u \in \Phi^{p,k}(\Omega)$ is called $p$–$k$–admissible if $u \in C^1(\Omega)$, $|Du|^{p-2}Du\in C^1(\Omega)$, and the spectrum of $DX$ lies in the convex Gårding cone $\Gamma_k = \{ \lambda \in \R^n : \sigma_j(\lambda) \geq 0,\ \forall 1 \leq j \leq k \}$. In the complex or Hermitian setting, the analogous construction involves the complex Hessian with eigenvalues of $(u_{i\bar j})$, and associated elementary symmetric functions [2512.07084]. ## 2. Critical Exponents, Solvability, and Liouville-Type Theorems For entire inequalities of the form \[ F_{k,p}[u](x) \geq (-u(x))^{\alpha} \quad \text{in } \R^n, \] the primary question concerns the existence of negative, $p$–$k$–admissible entire solutions. The main result asserts that the solvability is governed by a sharp lower-critical exponent \[ k_{p,*} = \frac{n(p-1)k}{n - pk}, \quad \text{for } pk < n. \] Negative solutions $u<0$ exist if and only if $\alpha > k_{p,*}$; for $\alpha \leq k_{p,*}$, no solution exists. The threshold $\alpha = k_{p,*}$ is sharp. The proof uses the construction of explicit radial barriers for $\alpha > k_{p,*}$ and integral identity techniques (based on integration by parts and local cut-offs) to derive nonexistence for $\alpha \leq k_{p,*}$ [2503.00895]. In the limit $k=1$ (the $p$–Laplacian), this reduces to the classical Serrin–Zou exponent; for $p=2$, one obtains the lower-critical exponent related to $k$-Hessian inequalities as studied by Phuc–Verbitsky and Ou. Further, more general $k$-Hessian type equations with gradient terms admit a Keller–Osserman-type integral criterion for global subsolutions, with a critical gradient parameter $\mu_0 = (n-k)/k$ for $k\ge2$ such that existence holds iff a certain integral diverges, extending classical results for the Laplacian and Hessian equations [2208.11103]. ## 3. Analytic and Geometric Structure: Admissibility, Symmetry, and Energy The $k$-Hessian operator $\sigma_k(D^2u)$ is elliptic on the cone $\mathcal{S}_k(\Omega) = \{ u \in C^2(\Omega): S_j(D^2u)\geq 0\ \forall j=1,\dots,k \}$. Such conditions are essential for both the analytic theory (maximum principle, regularity) and the geometry of $P$–$k$–Hessian flows. Energy functionals of the form \[ H_k(u;\Omega) = \frac{1}{k+1} \int_{\Omega} (-u)\,S_k(D^2u)\,dx \] and associated variational characterizations (for eigenvalues and torsional rigidity) play a central role, as does the symmetrization theory (Pólya–Szegö, Tso comparison, Faber–Krahn inequalities). Quantitative improvements involve deficit estimates in terms of the Hausdorff asymmetry, providing stability of isoperimetric-type inequalities [2407.20811]. In complex geometry, the definition is given in terms of $(\omega + dd^c u)^m \wedge \omega^{n-m}$, and m-Hessian capacity, energy, and singularity-type classes extend the framework to pluripotential theory and Kähler/Hermitian geometry [2512.07084, 2310.12400]. ## 4. Regularity, Estimates, and Existence Theory Regularity theory for $p$–$k$–Hessian and related equations is based on maximum principles, Evans–Krylov–Schauder regularity, and capacity-volume inequalities. Solutions admit $C^{2,\alpha}$ regularity in uniformly elliptic regimes ($k