Multiplicity of positive radial solutions for systems with mean curvature operator in Minkowski space

Zhiqian He; Liangying Miao; Zhiqian He; Liangying Miao

doi:10.3934/math.2021362

AIMS Mathematics

2021, Volume 6, Issue 6: 6171-6179. doi: 10.3934/math.2021362

Previous Article Next Article

Research article

Multiplicity of positive radial solutions for systems with mean curvature operator in Minkowski space

Zhiqian He ^{1
,
,},
Liangying Miao ²

1.
Department of Basic Teaching and Research, Qinghai University, Xining 810016, China
2.
School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, China

Received: 13 January 2021 Accepted: 29 March 2021 Published: 07 April 2021
MSC : 34B15, 35J66

In this paper, we are considered with the Dirichlet problem of quasilinear differential system with mean curvature operator in Minkowski space

$ \mathcal{M}(w): = \text{div}\Big(\frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\Big), $

in a ball in $ \mathbb{R}^N $. In particular, we deal with this system with Lane-Emden type nonlinearities in a superlinear case, by using the Leggett-Williams' fixed point theorem, we obtain the existence of three positive radial solutions.
- Minkowski curvature operator,
- system,
- positive radial solution,
- existence,
- Leggett-Williams' fixed point theorem
Citation: Zhiqian He, Liangying Miao. Multiplicity of positive radial solutions for systems with mean curvature operator in Minkowski space[J]. AIMS Mathematics, 2021, 6(6): 6171-6179. doi: 10.3934/math.2021362

Related Papers:

Abstract

In this paper, we are considered with the Dirichlet problem of quasilinear differential system with mean curvature operator in Minkowski space

$ \mathcal{M}(w): = \text{div}\Big(\frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\Big), $

in a ball in $ \mathbb{R}^N $. In particular, we deal with this system with Lane-Emden type nonlinearities in a superlinear case, by using the Leggett-Williams' fixed point theorem, we obtain the existence of three positive radial solutions.

References

[1]	R. Bartnik, L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys., 87 (1982), 131-152. doi: 10.1007/BF01211061
[2]	C. Bereanu, P. Jebelean, J. Mawhin, Radial solutions for systems involving mean curvature operators in Euclidean and Minkowski spaces, AIP Conference Proceedings, Melville: American Institute of Physics, 1124 (2009), 50-59.
[3]	C. Bereanu, P. Jebelean, J. Mawhin, Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, P. Am. Math. Soc., 137 (2009), 171-178.
[4]	C. Bereanu, P. Jebelean, P. J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264 (2013), 270-287. doi: 10.1016/j.jfa.2012.10.010
[5]	C. Bereanu, P. Jebelean, P. J. Torres, Multiple positive radial solutions for Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265 (2013), 644-659. doi: 10.1016/j.jfa.2013.04.006
[6]	C. Bereanu, P. Jebelean, J. Mawhin, The Dirichlet problem with mean curvature operator in Minkowski space-A variational approach, Adv. Nonlinear Stud., 14 (2014), 315-326.
[7]	A. Boscaggin, M. Garrione, Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball, Commun. Contemp. Math., 21 (2019), 1850006. doi: 10.1142/S0219199718500062
[8]	A. Boscaggin, G. Feltrin, Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefnite weight, Nonlinear Anal., 196 (2020), 111807. doi: 10.1016/j.na.2020.111807
[9]	X. F. Cao, G. W. Dai, N. Zhang, Global structure of positive solutions for problem with mean curvature operator on an annular domain, Rocky Mountain J. Math., 48 (2018), 1799-1814.
[10]	I. Coelho, C. Corsato, F. Obersnel, P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638.
[11]	I. Coelho, C. Corsato, S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Method. Nonlinear Anal., 44 (2014), 23-39.
[12]	G. W. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var., 55 (2016), 72. doi: 10.1007/s00526-016-1012-9
[13]	G. W. Dai, J. Wang, Nodal solutions to problem with mean curvature operator in Minkowski space, Differ. Integral Equ., 30 (2017), 463-480.
[14]	G. W. Dai, Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain, Indiana U. Math. J., 67 (2018), 2103-2121. doi: 10.1512/iumj.2018.67.7546
[15]	D. Gurban, P. Jebelean, Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane-Emden type nonlinearities, J. Differ. Equations, 266 (2019), 5377-5396. doi: 10.1016/j.jde.2018.10.030
[16]	D. Gurban, P. Jebelean, Positive radial solutions for systems with mean curvature operator in Minkowski space, Rend. Istit. Mat. Univ. Trieste, 49 (2017), 245-264.
[17]	D. Gurban, P. Jebelean, C. Serban, Nontrivial solutions for potential systems involving the mean curvature operator in Minkowski space, Adv. Nonlinear Stud., 17 (2017), 769-780. doi: 10.1515/ans-2016-6025
[18]	D. Gurban, P. Jebelean, C. Serban, Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space, Discrete Contin. Dyn. Syst., 40 (2020), 133-151. doi: 10.3934/dcds.2020006
[19]	Z. Q. He, L. Y. Miao, Uniqueness and multiplicity of positive solutions for one-dimensional prescribed mean curvature equation in Minkowski space, AIMS Mathematics, 5 (2020), 3840-3850. doi: 10.3934/math.2020249
[20]	S. C. Hu, H. Y. Wang, Convex solutions of boundary value problems arising from Monge-Ampère equations, Discret. Contin. Dyn. Syst., 16 (2006), 705-720. doi: 10.3934/dcds.2006.16.705
[21]	R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana U. Math. J., 28 (1979), 673-688. doi: 10.1512/iumj.1979.28.28046
[22]	Z. T. Liang, L. Duan, D. D. Ren, Multiplicity of positive radial solutions of singular Minkowski-curvature equations, Arch. Math., 113 (2019), 415-422. doi: 10.1007/s00013-019-01341-6
[23]	R. Y. Ma, H. L. Gao, Y. Q. Lu, Global structure of radial positive solutions for a prescribed mean curvature problem in a ball, J. Funct. Anal., 270 (2016), 2430-2455. doi: 10.1016/j.jfa.2016.01.020
[24]	R. Y. Ma, Positive solutions for Dirichlet problems involving the mean curvature operator in Minkowski space, Monatsh. Math., 187 (2018), 315-325. doi: 10.1007/s00605-017-1133-z
[25]	R. Y. Ma, Z. Q. He, Positive radial solutions for Dirichlet problem of quasilinear differential system with mean curvature operator in Minkowski space, J. Fixed Point Theory Appl., 23 (2021), 9. doi: 10.1007/s11784-020-00844-y
[26]	M. H. Pei, L. B. Wang, Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space, Appl. Math. Lett., 60 (2016), 50-55. doi: 10.1016/j.aml.2016.04.001
[27]	M. H. Pei, L. B. Wang, Positive radial solutions of a mean curvature equation in Minkowski space with strong singularity, P. Am. Math. Soc., 145 (2017), 4423-4430. doi: 10.1090/proc/13587
[28]	M. H. Pei, L. B. Wang, Positive radial solutions of a mean curvature equation in Lorentz-Minkowski space with strong singularity, Appl. Anal., 99 (2020), 1631-1637. doi: 10.1080/00036811.2018.1555322

Reader Comments

Your name:*

Email:*
© 2021 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)