Entire positive $ k $-convex solutions to $ k $-Hessian type equations and systems

Shuangshuang Bai; Xuemei Zhang; Meiqiang Feng; Shuangshuang Bai; Xuemei Zhang; Meiqiang Feng

doi:10.3934/era.2022025

Electronic Research Archive

2022, Volume 30, Issue 2: 481-491. doi: 10.3934/era.2022025

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Research article

Entire positive $ k $-convex solutions to $ k $-Hessian type equations and systems

1.
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
2.
School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China

Received: 15 December 2021 Revised: 14 January 2022 Accepted: 21 January 2022 Published: 08 February 2022

In this paper, we study the existence of entire positive solutions for the $ k $-Hessian type equation

$ {\rm S}_k(D^2u+\alpha I) = p(|x|)f^k(u), \ \ x\in \mathbb{R}^n $

and system

$ \begin{cases} {\rm S}_k(D^2u+\alpha I) = p(|x|)f^k(v), \ \ x\in \mathbb{R}^n, \\ {\rm S}_k(D^2v+\alpha I) = q(|x|)g^k(u), \ \ x\in \mathbb{R}^n, \end{cases} $

where $ D^2u $ is the Hessian of $ u $ and $ I $ denotes unit matrix. The arguments are based upon a new monotone iteration scheme.
- $ k $-Hessian type equation and system,
- entire positive $ k $-convex solution,
- monotone iterative,
- existence
Citation: Shuangshuang Bai, Xuemei Zhang, Meiqiang Feng. Entire positive $ k $-convex solutions to $ k $-Hessian type equations and systems[J]. Electronic Research Archive, 2022, 30(2): 481-491. doi: 10.3934/era.2022025

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Abstract

In this paper, we study the existence of entire positive solutions for the $ k $-Hessian type equation

$ {\rm S}_k(D^2u+\alpha I) = p(|x|)f^k(u), \ \ x\in \mathbb{R}^n $

and system

$ \begin{cases} {\rm S}_k(D^2u+\alpha I) = p(|x|)f^k(v), \ \ x\in \mathbb{R}^n, \\ {\rm S}_k(D^2v+\alpha I) = q(|x|)g^k(u), \ \ x\in \mathbb{R}^n, \end{cases} $

where $ D^2u $ is the Hessian of $ u $ and $ I $ denotes unit matrix. The arguments are based upon a new monotone iteration scheme.

References

[1]	J. I. E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana U. Math. J., 39 (1990), 355–382. https://doi.org/10.1512/iumj.1990.39.39020 doi: 10.1512/iumj.1990.39.39020
[2]	X. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana U. Math. J., 43 (1994), 25–54. https://doi.org/10.1512/iumj.1994.43.43002 doi: 10.1512/iumj.1994.43.43002
[3]	L. Caffarelli, Interior $W^{2, p}$ estimates for solutions of the Monge-Ampère equation, Ann. Math., 131 (1990), 135–150.
[4]	S. Cheng, S. Yau, On the regularity of the Monge-Ampère equation $det((\partial ^2u/\partial x_{i}\partial x_{j})) = F(x, u)$, Comm. Pure Appl. Math., 30 (1977), 41–68. https://doi.org/10.1002/cpa.3160300104 doi: 10.1002/cpa.3160300104
[5]	Z. Zhang, Large solutions to the Monge-Ampère equations with nonlinear gradient terms: Existence and boundary behavior, J. Differ. Equations, 264 (2018), 263–296. https://doi.org/10.1016/j.jde.2017.09.010 doi: 10.1016/j.jde.2017.09.010
[6]	W. Wei, Existence and multiplicity for negative solutions of $k$-Hessian equations, J. Differ. Equations, 263 (2017), 615–640. https://doi.org/10.1016/j.jde.2017.02.049 doi: 10.1016/j.jde.2017.02.049
[7]	X. Zhang, P. Xu, Y. Wu, The eigenvalue problem of a singular $k$-Hessian equation, Appl. Math. Lett., 124 (2022), 107666. https://doi.org/10.1016/j.aml.2021.107666 doi: 10.1016/j.aml.2021.107666
[8]	X. Zhang, J. Jiang, Y. Wu, B. Wiwatanapataphee, Iterative properties of solution for a general singular $n$-Hessian equation with decreasing nonlinearity, Appl. Math. Lett., 112 (2021), 106826. https://doi.org/10.1016/j.aml.2020.106826 doi: 10.1016/j.aml.2020.106826
[9]	X. Zhang, L. Liu, Y. Wu, Y. Cui, A sufficient and necessary condition of existence of blow-up radial solutions for a $k$-Hessian equation with a nonlinear operator, Nonlinear Anal.-Model., 25 (2020), 126–143. 10.15388/namc.2020.25.15736 doi: 10.15388/namc.2020.25.15736
[10]	L. Liu, 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
[11]	X. Zhang, J. Xu, J. Jiang, Y. Wu, Y. 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
[12]	X. Zhang, M. Feng, The existence and asymptotic behavior of boundary blow-up solutions to the $k$-Hessian equation, J. Differ. Equations, 267 (2019), 4626–4672. https://doi.org/10.1016/j.jde.2019.05.004 doi: 10.1016/j.jde.2019.05.004
[13]	X. Zhang, Y. Du, Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation, Calc. Var. Partial Differ. Equations, 57 (2018), 30. https://doi.org/10.1007/s00526-018-1312-3 doi: 10.1007/s00526-018-1312-3
[14]	X. Zhang, M. Feng, Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior, Adv. Nonlinear Anal., 9 (2020), 729–744. https://doi.org/10.1515/anona-2020-0023 doi: 10.1515/anona-2020-0023
[15]	M. Feng, X. Zhang, On a $k$-Hessian equation with a weakly superlinear nonlinearity and singular weights, Nonlinear Anal., 190 (2020), 111601. https://doi.org/10.1016/j.na.2019.111601 doi: 10.1016/j.na.2019.111601
[16]	J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure. Appl. Math., 10 (1957), 503–510. https://doi.org/10.1002/cpa.3160100402 doi: 10.1002/cpa.3160100402
[17]	R. Osserman, On the inequality $\Delta u \ge f(u)$, Pacific J. Math., 7 (1957), 1641–1647. https://doi.org/10.2140/pjm.1957.7.1641 doi: 10.2140/pjm.1957.7.1641
[18]	A.V. Lair, A.W. Wood, Large solutions of semilinear elliptic problems, Nonlinear Anal., 37 (1999), 805–812. https://doi.org/10.1016/S0362-546X(98)00074-1 doi: 10.1016/S0362-546X(98)00074-1
[19]	A.V. Lair, A. W. Wood, Existence of entire large positive solutions of semilinear elliptic systems, J. Differ. Equations, 164 (2000), 380–394. https://doi.org/10.1006/jdeq.2000.3768 doi: 10.1006/jdeq.2000.3768
[20]	L. Dupaigne, M. Ghergu, O. Goubet, G. Warnault, Entire large solutions for semilinear elliptic equations, J. Differ. Equations, 253 (2012), 2224–2251. https://doi.org/10.1016/j.jde.2012.05.024 doi: 10.1016/j.jde.2012.05.024
[21]	A. B. Dkhil, Positive solutions for nonlinear elliptic systems, Electron. J. Differ. Equations, 239 (2012), 1–10.
[22]	A.V. Lair, Entire large solutions to semilinear elliptic systems, J. Math. Anal. Appl., 382 (2011), 324–333. https://doi.org/10.1016/j.jmaa.2011.04.051 doi: 10.1016/j.jmaa.2011.04.051
[23]	H. Li, P. Zhang, Z. Zhang, A remark on the existence of entire positive solutions for a class of semilinear elliptic systems, J. Math. Anal. Appl., 365 (2010), 338–341. https://doi.org/10.1016/j.jmaa.2009.10.036 doi: 10.1016/j.jmaa.2009.10.036
[24]	Z. 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
[25]	Z. Zhang, H. Liu, Existence of entire radial large solutions for a class of Monge-Ampère type equations and systems, Rocky Mt., 2019. https://doi.org/10.1216/rmj.2020.50.1893
[26]	D. P. Covei, A remark on the existence of positive radial solutions to a Hessian system, AIMS Math., 6 (2021), 14035–14043. https://doi.org/10.3934/math.2021811 doi: 10.3934/math.2021811
[27]	L. Dai, Existence and nonexistence of subsolutions for augmented Hessian equations, Discrete Contin. Dyn. Syst., 40 (2020), 579–596. https://doi.org/10.3934/dcds.2020023 doi: 10.3934/dcds.2020023

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