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On a quadratic gradient natural term for the Pucci extremal operators

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla, Abiel Macedo

Abstract

We introduce a quadratic gradient type term for the Pucci extremal operators. Our analysis demonstrates that this proposed term extends the classical quadratic gradient term associated with the Laplace equation, and we investigate the impact of the Kazdan-Kramer transformation. As an application, we explore the existence, non-existence, uniqueness, Liouville-type results, and asymptotic behavior of solutions for the new class of Pucci equations under various conditions on both nonlinearity and domains.

On a quadratic gradient natural term for the Pucci extremal operators

Abstract

We introduce a quadratic gradient type term for the Pucci extremal operators. Our analysis demonstrates that this proposed term extends the classical quadratic gradient term associated with the Laplace equation, and we investigate the impact of the Kazdan-Kramer transformation. As an application, we explore the existence, non-existence, uniqueness, Liouville-type results, and asymptotic behavior of solutions for the new class of Pucci equations under various conditions on both nonlinearity and domains.

Paper Structure

This paper contains 6 sections, 10 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Sub- and superlinear growths
  3. Liouville type theorems and radially symmetric solutions
  4. Kazdan-Kramer change
  5. Existence for sub- and superlinear growths
  6. Liouville type results and radially symmetric solutions

Key Result

Theorem 1.3

Suppose the pair $f,g$ satisfies fg0 and fg_I. Then problem Pmais has a positive solution.

Theorems & Definitions (27)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 17 more