Opuscula Math. 46, no. 1 (2026), 55-72
https://doi.org/10.7494/OpMath.202512261
Opuscula Mathematica
A short note on Harnack inequality for k-Hessian equations with nonlinear gradient terms
Abstract. In this short note we study a Harnack inequality for \(k\)-Hessian equations that involve nonlinear lower-order terms which depend on the solution and its gradient.
Keywords: \(k\)-Hessian equation, \(k\)-convex solutions, Harnack inequality, Liouville property.
Mathematics Subject Classification: 35J60, 35J70, 35B45, 35B53.
- I. Birindelli, K.R. Payne, Principal eigenvalues for \(k\)-Hessian operators by maximum principle methods, Math. Eng. 3 (2021), no. 3, 1-37. https://doi.org/10.3934/mine.2021021
- L.A. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261-301. https://doi.org/10.1007/bf02392544
- K.-S. Chou, X.-J. Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math. 54 (2001), 1029-1064. https://doi.org/10.1002/cpa.1016
- F. Ferrari, A. Vitolo, Regularity properties for a class of non-uniformly elliptic Isaacs operators, Adv. Nonlinear Stud. 20 (2020), 213-241. https://doi.org/10.1515/ans-2019-2069
- X. Ji, J. Bao, Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc. 138 (2010), 175-188. https://doi.org/10.1090/s0002-9939-09-10032-1
- N.S. Trudinger, X.-J. Wang, Hessian measures I, Topol. Methods Nonlinear Anal. 10 (1997), 225-239. https://doi.org/10.12775/tmna.1997.030
- N.S. Trudinger, Weak solutions of Hessian equations, Comm. Partial Differential Equations 22 (1997), 1251-1261. https://doi.org/10.1080/03605309708821299
- A. Vitolo, Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues, Adv. Nonlinear Anal. 11 (2022), 1182-1200. https://doi.org/10.1515/anona-2022-0241
- X.-J. Wang, The \(k\)-Hessian equation, [in:] S.-Y. Chang, A. Ambrosetti, A. Malchiodi (eds), Geometric Analysis and PDEs, Lecture Notes in Mathematics, vol. 1977, Springer, Berlin, Heidelberg, 2009. https://doi.org/10.1007/978-3-642-01674-5_5
- Ahmed Mohammed (corresponding author)
https://orcid.org/0000-0001-5727-3614- Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA
- Giovanni Porru
- https://orcid.org/0000-0003-1207-5950
- Dipartimento di Matematica e Informatica, Università di Cagliari, 09124, Cagliari, Italy
- Communicated by Vicenţiu D. Rădulescu.
- Received: 2025-11-15.
- Revised: 2025-12-24.
- Accepted: 2025-12-26.
- Published online: 2026-01-27.
