Existence of radial solutions for $ k $-Hessian system

Hongliang Gao; Liyuan Wang; Jiemei Li; Hongliang Gao; Liyuan Wang; Jiemei Li

doi:10.3934/math.20231353

AIMS Mathematics

2023, Volume 8, Issue 11: 26498-26514. doi: 10.3934/math.20231353

Previous Article Next Article

Research article

Existence of radial solutions for $ k $-Hessian system

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China

Received: 25 June 2023 Revised: 04 September 2023 Accepted: 06 September 2023 Published: 15 September 2023
MSC : 35A01, 35A09, 35A24, 35A35

In this paper, we consider the existence of radial solutions to a $ k $-Hessian system in a general form. The existence of radial solutions is obtained under the assumptions that the nonlinearities in the given system satisfy $ k $-superlinear, $ k $-sublinear or $ k $-asymptotically linear at the origin and infinity, respectively. The results presented in this paper generalize some known results. Examples are given for the illustration of the main results.
- $ k $-Hessian system,
- Guo-Krasnosel'skii fixed point theorem,
- existence,
- cone
Citation: Hongliang Gao, Liyuan Wang, Jiemei Li. Existence of radial solutions for $ k $-Hessian system[J]. AIMS Mathematics, 2023, 8(11): 26498-26514. doi: 10.3934/math.20231353

Related Papers:

Abstract

In this paper, we consider the existence of radial solutions to a $ k $-Hessian system in a general form. The existence of radial solutions is obtained under the assumptions that the nonlinearities in the given system satisfy $ k $-superlinear, $ k $-sublinear or $ k $-asymptotically linear at the origin and infinity, respectively. The results presented in this paper generalize some known results. Examples are given for the illustration of the main results.

References

[1]	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. http://doi.org/10.1007/bf02392544 doi: 10.1007/bf02392544
[2]	X. J. Wang, The $k$-Hessian equation, In: Geometric analysis and PDEs, Heidelberg: Springer, 1977 (2009), 177–252. http://doi.org/10.1007/978-3-642-01674-5_5
[3]	Z. Zhang, K. Wang, Existence and non-existence of solutions for a class of Monge-Ampère equations, J. Differential Equations, 246 (2009), 2849–2875. https://doi.org/10.1016/j.jde.2009.01.004 doi: 10.1016/j.jde.2009.01.004
[4]	J. Bao, H. Li, L. Zhang, Monge-Ampère equation on exterior domains, Calc. Var. Partial Differential Equations, 52 (2015), 39–63. https://doi.org/10.1007/s00526-013-0704-7 doi: 10.1007/s00526-013-0704-7
[5]	D. P. Covei, Solutions with radial symmetry for a semilinear elliptic system with weights, Appl. Math. Lett., 76 (2018), 187–194. https://doi.org/10.1016/j.aml.2017.09.003 doi: 10.1016/j.aml.2017.09.003
[6]	Z. J. Zhang, Optimal global and boundary asymptotic behavior of large solutions to the Monge-Ampère equation, J. Funct. Anal., 278 (2020), 108512. https://doi.org/10.1016/j.jfa.2020.108512 doi: 10.1016/j.jfa.2020.108512
[7]	H. Wang, Convex solutions of systems arising from Monge-Ampère equations, Electron. J. Qual. Theory Differ. Equ., Spec. Ed. I, 26 (2009), 1–8. https://doi.org/10.14232/ejqtde.2009.4.26 doi: 10.14232/ejqtde.2009.4.26
[8]	B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations, 8 (1999), 45–69. https://doi.org/10.1007/s005260050116 doi: 10.1007/s005260050116
[9]	G. W. Dai, Bifurcation and admissible solutions for the Hessian equation, J. Funct. Anal., 273 (2017), 3200–3240. https://doi.org/10.1016/j.jfa.2017.08.001 doi: 10.1016/j.jfa.2017.08.001
[10]	S. Chang, M. J. Gursky, P. C. Yang, An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. Math., 155 (2004), 709–787. https://doi.org/10.2307/3062131 doi: 10.2307/3062131
[11]	Y. C. Shiu, T. Y. Shing, On the regularity of the solution of the n-dimensional Minkowski problem, Commun. Pure Appl. Math., 29 (1976), 495–516. https://doi.org/10.1002/cpa.3160290504 doi: 10.1002/cpa.3160290504
[12]	X. J. Chen, Q. Tu, N. Xiang, A class of Hessian quotient equations in Euclidean space, J. Differential Equations, 269 (2020), 11172–11194. https://doi.org/10.1016/j.jde.2020.08.048 doi: 10.1016/j.jde.2020.08.048
[13]	Z. J. Zhang, S. Zhou, Existence of entire positive $k$-convex radial solutions to Hessian equations and systems with weights, Appl. Math. Lett., 50 (2015), 48–55. https://doi.org/10.1016/j.aml.2015.05.018 doi: 10.1016/j.aml.2015.05.018
[14]	D. Covei, A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights, Acta Math. Sci., 37 (2017), 47–57. https://doi.org/10.1016/S0252-9602(16)30114-X doi: 10.1016/S0252-9602(16)30114-X
[15]	G. T. Wang, Z. D. Yang, L. H. Zhang, D. Baleanu, Radial solutions of a nonlinear $k$-Hessian system involving a nonlinear operator, Commun. Nonlinear Sci., 91 (2020), 105396. https://doi.org/10.1016/j.cnsns.2020.105396 doi: 10.1016/j.cnsns.2020.105396
[16]	X. G. Zhang, H. Tain, Y. H. Wu, B. Wiwatanapataphee, The radial solution for an eigenvalue problem of singular augmented Hessian equation, Appl. Math. Lett., 134 (2022), 108330. https://doi.org/10.1016/j.aml.2022.108330 doi: 10.1016/j.aml.2022.108330
[17]	X. G. Zhang, P. T. Xu, Y. H. Wu, The eigenvalue problem of a singular $k$ -Hessian equation, Appl. Math. Lett., 124 (2021), 107666. https://doi.org/10.1016/j.aml.2021.107666 doi: 10.1016/j.aml.2021.107666
[18]	M. Q. Feng, New results of coupled system of $k$-Hessian equations, Appl. Math. Lett., 94 (2019), 196–203. https://doi.org/10.1016/j.aml.2019.03.008 doi: 10.1016/j.aml.2019.03.008
[19]	X. M. Zhang, Analysis of nontrivial radial solutions for singular superlinear $k$-Hessian equations, Appl. Math. Lett., 106 (2020), 106409. https://doi.org/10.1016/j.aml.2020.106409 doi: 10.1016/j.aml.2020.106409
[20]	J. X. He, X. G. Zhang, L. S. Liu, Y. H. Wu, Existence and nonexistence of radial solutions of Dirichlet problem for a class of general $k$-Hessian equations, Nonlinear Anal. Model., 23 (2018), 475–492. https://doi.org/10.15388/NA.2018.4.2 doi: 10.15388/NA.2018.4.2
[21]	Z. D. Yang, Z. B. Bai, Existence and multiplicity of radial solutions for a $k$-Hessian system, J. Math. Anal. Appl., 512 (2022), 126159. https://doi.org/10.1016/j.jmaa.2022.126159 doi: 10.1016/j.jmaa.2022.126159
[22]	A. V. Lair, A. Mohammed, Large solutions to semi-linear elliptic systems with variable exponents, J. Math. Anal. Appl., 420 (2014), 1478–1499. https://doi.org/10.1016/j.jmaa.2014.06.068 doi: 10.1016/j.jmaa.2014.06.068
[23]	Z. Zhang, Large solutions to the Monge-Ampère equations with nonlinear gradient terms: Existence and boundary behavior, J. Differential Equations, 264 (2018), 263–296. https://doi.org/10.1016/j.jde.2017.09.010 doi: 10.1016/j.jde.2017.09.010
[24]	X. G. Zhang, J. F. Xu, J. Q. Jiang, Y. H. Wu, Y. J. Cui, The convergence analysis and uniqueness of blow-up solutions for a Dirichlet problem of the general $k$-Hessian equations, Appl. Math. Lett., 102 (2020), 106124. https://doi.org/10.1016/j.aml.2019.106124 doi: 10.1016/j.aml.2019.106124
[25]	X. G. Zhang, P. Chen, Y. H. Wu, B. Wiwatanapataphee, A necessary and sufficient condition for the existence of entire large solutions to a $k$-Hessian system, Appl. Math. Lett., 145 (2023), 108745. https://doi.org/10.1016/j.aml.2023.108745 doi: 10.1016/j.aml.2023.108745
[26]	Z. B. Bai, Z. D. Yang, Existence of $k$-convex solutions for the $k$-Hessian equation, Mediterr. J. Math., 20 (2023), 150. https://doi.org/10.1007/s00009-023-02364-8 doi: 10.1007/s00009-023-02364-8
[27]	X. H. Ji, J. G. Bao, Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc., 138 (2010), 175–188. http://doi.org/10.1090/S0002-9939-09-10032-1 doi: 10.1090/S0002-9939-09-10032-1
[28]	D. J. Guo, V. Lakshmikantham, Positive fixed point theory, In: Nonlinear problems in abstract cones, New York: Academic Press, 1988. https://doi.org/10.1016/B978-0-12-293475-9.50005-4

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)