On Liouville-type theorems for 𝑘-Hessian equations with gradient terms

Doerr, Cameron; Mohammed, Ahmed

doi:10.1090/proc/16737

On Liouville-type theorems for $k$-Hessian equations with gradient terms
HTML articles powered by AMS MathViewer

by Cameron Doerr and Ahmed Mohammed

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/16737

Published electronically: April 15, 2024

HTML | PDF | Request permission

Abstract:

In this paper, we investigate several Liouville-type theorems related to $k$-Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to $S_k[D^2u]\ge h(u,|Du|)$ in $\mathbb {R}^n$. The results depend on some qualified growth conditions of $h$ at infinity. A Liouville-type result to subsolutions of a prototype equation $S_k[D^2u]=f(u)+g(u)\varpi (|Du|)$ is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to $S_k[D^2u]=f(u)+g(u)|Du|^q$ for $0\le q<k+1$ is also given.

References

Tilak Bhattacharya and Ahmed Mohammed, Maximum principles for $k$-Hessian equations with lower order terms on unbounded domains, J. Geom. Anal. 31 (2021), no. 4, 3820–3862. MR 4236544, DOI 10.1007/s12220-020-00415-0
Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
Isabeau Birindelli, Giulio Galise, and Hitoshi Ishii, A family of degenerate elliptic operators: maximum principle and its consequences, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 2, 417–441. MR 3765548, DOI 10.1016/j.anihpc.2017.05.003
I. Birindelli, G. Galise, and H. Ishii, Existence through convexity for the truncated Laplacians, Math. Ann. 379 (2021), no. 3-4, 909–950. MR 4238256, DOI 10.1007/s00208-019-01953-x
Isabeau Birindelli, Giulio Galise, and Fabiana Leoni, Liouville theorems for a family of very degenerate elliptic nonlinear operators, Nonlinear Anal. 161 (2017), 198–211. MR 3673001, DOI 10.1016/j.na.2017.06.002
Alessandra Cutrì and Fabiana Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 2, 219–245 (English, with English and French summaries). MR 1753094, DOI 10.1016/S0294-1449(00)00109-8
Luis Caffarelli, Yan Yan Li, and Louis Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. I, J. Fixed Point Theory Appl. 5 (2009), no. 2, 353–395. MR 2529505, DOI 10.1007/s11784-009-0107-8
L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), no. 3-4, 261–301. MR 806416, DOI 10.1007/BF02392544
Luis Caffarelli, Yanyan Li, and Louis Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators, Comm. Pure Appl. Math. 66 (2013), no. 1, 109–143. MR 2994551, DOI 10.1002/cpa.21412
Italo Capuzzo Dolcetta, Fabiana Leoni, and Antonio Vitolo, Entire subsolutions of fully nonlinear degenerate elliptic equations, Bull. Inst. Math. Acad. Sin. (N.S.) 9 (2014), no. 2, 147–161. MR 3237064
Italo Capuzzo Dolcetta, Fabiana Leoni, and Antonio Vitolo, On the inequality $F(x,D^2u)\geq f(u)+g(u)|Du|^q$, Math. Ann. 365 (2016), no. 1-2, 423–448. MR 3498917, DOI 10.1007/s00208-015-1280-2
Gregorio Díaz, A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller-Osserman condition, Math. Ann. 353 (2012), no. 1, 145–159. MR 2910785, DOI 10.1007/s00208-011-0678-8
Patricio Felmer, Alexander Quaas, and Boyan Sirakov, Solvability of nonlinear elliptic equations with gradient terms, J. Differential Equations 254 (2013), no. 11, 4327–4346. MR 3035435, DOI 10.1016/j.jde.2013.03.003
G. Galise, S. Koike, O. Ley, and A. Vitolo, Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term, J. Math. Anal. Appl. 441 (2016), no. 1, 194–210. MR 3488054, DOI 10.1016/j.jmaa.2016.03.083
Fausto Ferrari and Antonio Vitolo, Regularity properties for a class of non-uniformly elliptic Isaacs operators, Adv. Nonlinear Stud. 20 (2020), no. 1, 213–241. MR 4054947, DOI 10.1515/ans-2019-2069
Xiaohu Ji and Jiguang Bao, Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc. 138 (2010), no. 1, 175–188. MR 2550182, DOI 10.1090/S0002-9939-09-10032-1
J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math. 10 (1957), 503–510. MR 91407, DOI 10.1002/cpa.3160100402
Robert Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math. 7 (1957), 1641–1647. MR 98239
Antonio Vitolo, Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues, Adv. Nonlinear Anal. 11 (2022), no. 1, 1182–1200. MR 4390824, DOI 10.1515/anona-2022-0241
A. Vitolo, Singular elliptic equations with directional diffusion, Math. Eng. 3 (2021), no. 3, Paper no. 027, 16 pp.
Antonio Vitolo, Existence of positive entire solutions of fully nonlinear elliptic equations, J. Elliptic Parabol. Equ. 4 (2018), no. 2, 293–304. MR 3905635, DOI 10.1007/s41808-018-0019-0
Xu-Jia Wang, The $k$-Hessian equation, Geometric analysis and PDEs, Lecture Notes in Math., vol. 1977, Springer, Dordrecht, 2009, pp. 177–252. MR 2500526, DOI 10.1007/978-3-642-01674-5_{5}