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A Gradient Type Term for the k-Hessian Equation

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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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Acknowledgements

The authors would like to thank the anonymous referee for valuable suggestions that improved the quality of the paper.

Funding

The work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) Grant 309491/2021-5 and Fundação de Amparo à Pesquisa do Estado do Piauí - FAPEPI.

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  1. Mykael Cardoso, Jefferson de Brito Sousa, and José Francisco de Oliveira have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Federal University of Piauí, Teresina, Piauí, 64049-550, Brazil

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

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  1. Mykael Cardoso
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  2. Jefferson de Brito Sousa
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  3. José Francisco de Oliveira
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All authors contributed equally to this work.

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Correspondence to José Francisco de Oliveira.

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Cardoso, M., de Brito Sousa, J. & de Oliveira, J.F. A Gradient Type Term for the k-Hessian Equation. J Geom Anal 34, 19 (2024). https://doi.org/10.1007/s12220-023-01458-9

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