P-k-Hessian Equations in Nonlinear Analysis

• P–k–Hessian equations are nonlinear PDEs defined by combining gradient- and Hessian-dependent nonlinearities, generalizing both the p-Laplacian and k–Hessian operators.
• They exhibit sharp critical exponents and rely on methods like radial barriers and variational techniques to establish existence or nonexistence of solutions.
• These equations underpin geometric flows, regularity theory, and numerical methods, bridging classical PDEs with modern pluripotential and complex geometric analyses.

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 (Gao et al., 2 Mar 2025).

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 (Pang et al., 8 Dec 2025).

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 $\alpha \leq k_{p,*}$ (Gao et al., 2 Mar 2025).

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 (Ji et al., 2022).

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 (Masiello et al., 2024).

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 (Pang et al., 8 Dec 2025, Lin, 2023).

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 ($k<n$ in the complex case), whereas degenerate or critical cases ($k=n$) yield only $C^{1,\alpha}$ a priori (Li, 2024).

Existence theorems for degenerate complex Hessian equations utilize pluripotential techniques, including $L^{\infty}$ estimates (via capacity-volume decay and De Giorgi iteration), comparison principles, and the construction of global solutions with prescribed singularities or within specific cohomology or energy classes. On compact Hermitian or Kähler manifolds, the solvability reduces to positivity and big/nef class conditions (Pang et al., 8 Dec 2025, Chen, 2021), extending Bedford–Taylor's theory to $(\omega, m)$-Hessian contexts.

For parabolic $k$–Hessian evolution equations $-u_t \sigma_k(D^2u) = \psi(x,t,u)$, sharp Pogorelov and Liouville-type estimates control derivatives and classify entire solutions: under suitable convexity and monotonicity, solutions must be quadratic in $x$ and affine in $t$ (He et al., 2019, Bao et al., 2022).

5. Numerics, Iterative Methods, and Applications

Computational approaches for these nonlinear elliptic and parabolic PDEs rely on finite-difference discretizations (e.g., 9-point stencils), iterative solvers (Newton-type, subharmonicity-preserving, and Gauss–Seidel schemes), and careful handling of viscosity/weak solutions and admissibility constraints. Local quadratic convergence holds for smooth non-degenerate solutions, but special strategies (subharmonicity-preserving) are used in presence of degeneracies and low regularity (Awanou, 2014).

Applications are found in geometric flows (J-flow, Lagrangian mean curvature flow), extremal metrics, nonlinear elasticity, geometric optics, and convex geometry. Analytical features such as mixed Hodge-index inequalities (via Gårding hyperbolic polynomials) tie these PDEs to deep geometric-analytic and intersection-theoretic results (Xiao, 2018).

6. Extensions: Higher Order, Fractional, Complex, and Pluripotential Equations

Higher-order (polyharmonic) $k$-Hessian systems involve compositions such as $(-\Delta)^m u = S_k[-u] + \lambda f$, with existence theory obtained via critical Sobolev embedding, weak continuity, and fixed-point arguments. Nonlocal (fractional) analogues replace $(-\Delta)^m$ with powers of $(-\Delta)^{\alpha}$ (Balodis et al., 2016).

In Hermitian and Kähler geometry, Hessian type equations extend to the degenerate setting with $(\omega, m)$-positivity, big classes, nef Bott–Chern classes, and to equations with mild analytic singularities or prescribed logarithmic poles, with existence and regularity dictated by pluripotential and capacity theories (Pang et al., 8 Dec 2025, Lin, 2023). Krylov-type and Hessian quotient equations generalize this structure to combinations and ratios of symmetric polynomials under suitable cone constraints (Chen, 2021).

7. Structural and Theoretical Unification

The $p$–$k$–Hessian theory encompasses and extends several classical nonlinear PDEs—the $p$–Laplacian, $k$–Hessian, Monge–Ampère, and their complex analogues—via the unified language of elementary symmetric polynomials, Gårding cones, and hyperbolic polynomials. The fundamental analytic properties stem from concavity, ellipticity, and symmetrization, enabling both rigidity phenomena (Liouville theorems) and quantitative stability in isoperimetric, eigenvalue, and torsional rigidity inequalities.

The same mathematical structure controls:

• Existence and nonexistence thresholds (critical exponents)
• A priori regularity and maximum principles
• Quantitative isoperimetric inequalities and variational characterizations
• Cohomological and energy-based criteria for solvability in geometric contexts

Consequently, P–k–Hessian type equations serve as a keystone in the interplay of nonlinear analysis, geometry, and modern pluripotential theory, with ongoing research addressing open problems in regularity, singularities, multiplicity, and nonlocal extensions (Gao et al., 2 Mar 2025, Pang et al., 8 Dec 2025, Li, 2024, Awanou, 2014, Balodis et al., 2016, Chen, 2021).