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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data availability is not applicable to this work.
Code Availability
Not applicable.
References
Abdellaoui, B., Dall’Aglio, A., Peral, I.: Some remarks on elliptic problems with critical growth in the gradient. J. Differ. Equ. 222(1), 21–62 (2006)
Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second elliptic equations, I. Monge-Ampere equations. Commun. Pure Appl. Math. 37, 369–402 (1984)
Chou, K.S., Wang, X.-J.: A variational theory of the Hessian equation. Commun. Pure Appl. Math. 54, 1029–1064 (2001)
De Figueiredo, D.G., Miyagaki, O.H., Ruf, B.: Elliptic equations in \(\mathbb{R} ^2\) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3, 139–153 (1995)
De Figueiredo, D.G., Gossez, J.P., Quoirin, H.R., Ubilla, P.: Elliptic equations involving the \(p\)-Laplacian and a gradient term having natural growth. Rev. Mat. Iberoam. 35, 173–194 (2019)
De Oliveira, J.F., Ubilla, P.: Admissible solutions to Hessian equations with exponential growth. Rev. Mat. Iberoam. 37, 747–773 (2021)
De Oliveira, J.F., Do Ó, J.M., Ubilla, P.: Existence for a \(k\)-Hessian equation involving supercritical growth. J. Differ. Equ. 267, 1001–1024 (2019)
García-Melían, J., Iturriaga, L., Quoirin, H.R.: A priori bounds and existence of solutions for slightly superlinear elliptic problems. Adv. Nonlinear Stud. 15(4), 923–938 (2015)
Guan, B.: The Dirichlet problem for Hessian equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 8, 45–69 (1999)
Guowei, D.: Bifurcation and admissible solutions for the Hessian equation. J. Funct. Anal. 273, 3200–3240 (2017)
Hamid, H.A., Bidaut-Veron, M.F.: On the connection between two quasilinear elliptic problems with source terms of order \(0\) or \(1\). Commun. Contemp. Math. 12(5), 727–788 (2010)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, New York (2013)
Jeanjean, L., Quoirin, H.: R Multiple solutions for an indefinite elliptic problem with critical growth in the gradient. Proc. Am. Math. Soc. 144(2), 575–586 (2016)
Jeanjean, L., Sirakov, B.: Existence and multiplicity for elliptic problems with quadratic growth in the gradient. Commun. Partial Differ. Equ. 38(2), 244–264 (2013)
Kazdan, J.L., Kramer, R.J.: Invariant criteria for existence of solutions to second-order quasilinear elliptic equations. Commun. Pure Appl. Math. 31(5), 619–645 (1978)
Li, J., Yin, J., Ke, Y.: Existence of positive solutions for the p-Laplacian with p-gradient term. J. Math. Anal. Appl. 383(1), 147–158 (2011)
Phuc, N.C., Verbisky, I.E.: Quasilinear and Hessian equations of Lane-Emden type. Ann. Math. 168, 859–914 (2008)
Phuc, N.C., Verbisky, I.E.: Singular quasilinear and Hessian equations and inequalities. J. Funct. Anal. 256, 1875–1906 (2009)
Porretta, A., Segura de Leon, S.: Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. (9) 85(3), 465–492 (2006)
Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35, 681–703 (1986)
Sheng, W.M., Trudinger, N.S., Wang, X.J.: The Yamabe problem for higher order curvatures. J. Differ. Geom. 77, 515–553 (2007)
Tian, G.-T., Wang, X.-J.: Moser-Trudinger type inequalities for the Hessian equation. J. Funct. Anal. 259(8), 1974–2002 (2010)
Trudinger, N.S., Wang, X.-J.: Hessian measures II. Ann. Math. 150, 579–604 (1999)
Tso, K.: Remarks on critical exponents for Hessian operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 113–122 (1990)
Wang, X.-J.: A class of fully nonlinear elliptic equations and related functionals. Indiana Univ. Math. J. 43(1), 25–54 (1994)
Wang, X.-J.: The k-Hessian Equation. Lecture Notes in Mathematics (1977), pp. 177–252. Springer, New York (2009)
Wei, W.: Uniqueness theorems for negative radial solutions of k-Hessian equations in a ball. J. Differ. Equ. 261, 3756–3771 (2016)
Wei, W.: Existence and multiplicity for negative solutions of k-Hessian equations. J. Differ. Equ. 263, 615–640 (2017)
Zhang, X., Feng, M.: The existence and asymptotic behavior of boundary blow-up solutions to the k-Hessian equation. J. Differ. Equ. 267, 4626–4672 (2019)
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.
Author information
Authors and Affiliations
Contributions
All authors contributed equally to this work.
Corresponding author
Ethics declarations
Conflict of interest
All authors declare that there is no Conflict of interest.
Ethical Approval
Not applicable.
Consent to Participate
All authors contributed equally to this work of their own free will.
Consent for Publication
All authors read and approved the final manuscript for publication.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01458-9