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arXiv · 2301.07201

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

Mykael de Araújo Cardoso·Jefferson de Brito Sousa·José Francisco de Oliveira
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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.

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Mykael de Araújo Cardoso, Jefferson de Brito Sousa, José Francisco de Oliveira. 2023-01-17. A gradient type term for the $k$-Hessian equation. https://doi.org/10.1007/s12220-023-01458-9

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