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On a class of Cauchy problems with applications in nonlinear partial differential equations

Feida Jiang, Neil S. Trudinger, Qiao-Qiao Xu

TL;DR

The paper develops a unified ODE framework for a broad Cauchy problem on $(0,+\infty)$ and proves a sharp Keller–Osserman criterion governing the existence of entire solutions. It then applies these results to nonlinear PDEs involving $k$-Hessian and $\Pi_k$-Hessian operators, including degenerate and singular right-hand sides and generalized $p$-Hessian variants, establishing necessary and sufficient KO-type conditions across multiple parameter regimes. A key feature is handling the critical ranges of $\tau$ relative to $q$ and $\theta$, notably the novel $0<\theta<1$ cases and the $p$-generalized Hessian scenarios, thereby yielding a comprehensive set of criteria for entire solvability and subsolution existence in fully nonlinear elliptic equations in $\mathbb{R}^n$. The results provide a robust toolkit for constructing entire subsolutions and understanding global solvability in elliptic PDEs with geometric operators, with duality between $k$-Hessian and $\Pi_k$-Hessian frameworks. These contributions extend classical Keller–Osserman theory to a broad class of nonlinear, potentially degenerate equations, offering implications for geometric problems and nonlinear PDE modeling.

Abstract

In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of nonlinear partial differential equations in n-dimensional Euclidean space. As applications of the general framework, we present such results for two series of nonlinear equations: a series of k-Hessian type equations and a new series of P-k-Hessian type equations. These results are also proved for the more general p-Hessian matrices, with p > 1, and degenerate non homogeneous terms.

On a class of Cauchy problems with applications in nonlinear partial differential equations

TL;DR

The paper develops a unified ODE framework for a broad Cauchy problem on and proves a sharp Keller–Osserman criterion governing the existence of entire solutions. It then applies these results to nonlinear PDEs involving -Hessian and -Hessian operators, including degenerate and singular right-hand sides and generalized -Hessian variants, establishing necessary and sufficient KO-type conditions across multiple parameter regimes. A key feature is handling the critical ranges of relative to and , notably the novel cases and the -generalized Hessian scenarios, thereby yielding a comprehensive set of criteria for entire solvability and subsolution existence in fully nonlinear elliptic equations in . The results provide a robust toolkit for constructing entire subsolutions and understanding global solvability in elliptic PDEs with geometric operators, with duality between -Hessian and -Hessian frameworks. These contributions extend classical Keller–Osserman theory to a broad class of nonlinear, potentially degenerate equations, offering implications for geometric problems and nonlinear PDE modeling.

Abstract

In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of nonlinear partial differential equations in n-dimensional Euclidean space. As applications of the general framework, we present such results for two series of nonlinear equations: a series of k-Hessian type equations and a new series of P-k-Hessian type equations. These results are also proved for the more general p-Hessian matrices, with p > 1, and degenerate non homogeneous terms.
Paper Structure (12 sections, 13 theorems, 143 equations, 1 table)

This paper contains 12 sections, 13 theorems, 143 equations, 1 table.

Table of Contents

  1. Introduction
  2. Local existence and regularity of the Cauchy problem
  3. Entire existence and nonexistence of the Cauchy problem
  4. Applications to $k$-Hessian type equations
  5. Standard $k$-Hessian case
  6. Extensions to $p$-$k$ Hessian equations
  7. Keller-Osserman theorem
  8. Applications to $\Pi_k$-Hessian type equations
  9. Standard $\Pi_k$-Hessian case
  10. Extensions to $p$-$\Pi_k$ Hessian equations
  11. Keller-Osserman theorem
  12. Remarks and Examples

Key Result

Theorem 1.1

Assume that $g: \mathbb{R} \to (0,+\infty)$ is a non-decreasing continuous function, the constants $q\ge 0$, $\theta> 0$, $C>0$ and $\tau > q\theta$. Then for any constant $a$, there exists a positive number $R$ such that the Cauchy problem 1.1 admits a solution $v \in C^1[0, R)\cap C^2(0, R)$ with

Theorems & Definitions (45)

  • Theorem 1.1: Local existence and regularity
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • proof : Proof of Theorem \ref{['Th 1.1']}
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 35 more