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A gradient type term for the $k$-Hessian equation

Mykael de Araújo Cardoso, Jefferson de Brito Sousa, José Francisco de Oliveira

Abstract

In this paper, we propose a gradient type term for the $k$-Hessian equation that extends for $k>1$ the classical quadratic gradient term associated with the Laplace equation. We prove that such as gradient term is invariant by the Kazdan-Kramer change of variables. As applications, we ensure the existence of solutions for a new class of $k$-Hessian equation in the sublinear and superlinear cases for Sobolev type growth. The threshold for existence is obtained in some particular cases. In addition, for the Trudinger-Moser type growth regime, we also prove the existence of solutions under either subcritical or critical conditions.

A gradient type term for the $k$-Hessian equation

Abstract

In this paper, we propose a gradient type term for the -Hessian equation that extends for the classical quadratic gradient term associated with the Laplace equation. We prove that such as gradient term is invariant by the Kazdan-Kramer change of variables. As applications, we ensure the existence of solutions for a new class of -Hessian equation in the sublinear and superlinear cases for Sobolev type growth. The threshold for existence is obtained in some particular cases. In addition, for the Trudinger-Moser type growth regime, we also prove the existence of solutions under either subcritical or critical conditions.
Paper Structure (11 sections, 15 theorems, 71 equations)

This paper contains 11 sections, 15 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Statements of results
  3. The Sobolev case
  4. The Trudinger-Moser case
  5. The change of variable of Kazdan-Kramer type for k-Hessian equation
  6. Existence for the Sobolev case: Proof of Theorem \ref{['thm2-sup']} and Theorem \ref{['thm1-sub']}
  7. Non-existence results: Proof of Theorem \ref{['non-ex']}
  8. Proof of Theorem \ref{['non-ex']}.
  9. Existence for the Trudinger-Moser case
  10. Case 1: Subcritical.
  11. Case 2: Critical.

Key Result

Theorem 2.1

Suppose $f$ and $g$ satisfying $(H_f)$, $(H_g)$ and such that the function $h$ given by h-transformed belongs to $C^{1,1}(\overline{\Omega \times \mathbb{R}^{-}})$. In addition, assume f-superlinear, $(H_{SC})$, $(H_{AR})$ and the following Then, the problem equação 1.1 has a nontrivial solution $u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$.

Theorems & Definitions (27)

  • Theorem 2.1: superlinear case
  • Theorem 2.2: sublinear case
  • Corollary 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6: subcritical case
  • Theorem 2.7: critical case
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 17 more