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

TL;DR

This paper introduces a new gradient term for the $k$-Hessian equation, extending classical concepts, and proves existence results for solutions under various growth conditions, including Sobolev and Trudinger-Moser regimes.

Contribution

It proposes a novel gradient term invariant under specific transformations and establishes existence results for a broad class of $k$-Hessian equations.

Findings

Existence of solutions in sublinear and superlinear cases.

Existence under Trudinger-Moser growth conditions.

Thresholds for solution existence in particular cases.

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.