Bifurcation curves of positive solutions for one-dimensional Minkowski curvature problem

Zhiqian He; Man Xu; Yanzhong Zhao; Xiaobin Yao; Zhiqian He; Man Xu; Yanzhong Zhao; Xiaobin Yao

doi:10.3934/math.2022934

AIMS Mathematics

2022, Volume 7, Issue 9: 17001-17018. doi: 10.3934/math.2022934

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Research article

Bifurcation curves of positive solutions for one-dimensional Minkowski curvature problem

1.
School of Mathematics and Physics, Qinghai University, Xining 810016, China
2.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
3.
School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China

Received: 19 May 2022 Revised: 29 June 2022 Accepted: 04 July 2022 Published: 19 July 2022
MSC : 34C23, 35J60, 34B18

In this paper, we study the shape of the bifurcation curves of positive solutions for the one-dimensional Minkowski-curvature problem. By developing some new time mapping techniques, we find that the bifurcation curve is $ \subset $-shaped/monotone increasing/$ S $-like shaped on the $ (\lambda, ||u||_\infty) $ plane when the nonlinearity satisfies different assumptions. Finally, two examples are given to illustrate our result.
- mean curvature equation,
- positive solutions,
- bifurcation curve,
- time mapping,
- existence
Citation: Zhiqian He, Man Xu, Yanzhong Zhao, Xiaobin Yao. Bifurcation curves of positive solutions for one-dimensional Minkowski curvature problem[J]. AIMS Mathematics, 2022, 7(9): 17001-17018. doi: 10.3934/math.2022934

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Abstract

In this paper, we study the shape of the bifurcation curves of positive solutions for the one-dimensional Minkowski-curvature problem. By developing some new time mapping techniques, we find that the bifurcation curve is $ \subset $-shaped/monotone increasing/$ S $-like shaped on the $ (\lambda, ||u||_\infty) $ plane when the nonlinearity satisfies different assumptions. Finally, two examples are given to illustrate our result.

References

[1]	A. Ambrosetti, P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl., 73 (1980), 411–422. https://doi.org/10.1016/0022-247X(80)90287-5 doi: 10.1016/0022-247X(80)90287-5
[2]	C. Bereanu, P. Jebelean, J. Mawhin, Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, Calc. Var. Partial Differ. Equ., 46 (2013), 113–122. https://doi.org/10.1007/s00526-011-0476-x doi: 10.1007/s00526-011-0476-x
[3]	A. Boscaggin, G. Feltrin, Positive periodic solutions to an indefinite Minkowski-curvature equation, J. Differ. Equ., 269 (2020), 5595–5645. https://doi.org/10.1016/j.jde.2020.04.009 doi: 10.1016/j.jde.2020.04.009
[4]	A. Boscaggin, G. Feltrin, Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefinite weight, Nonlinear Anal., 196 (2020), 1–14. https://doi.org/10.1016/j.na.2020.111807 doi: 10.1016/j.na.2020.111807
[5]	K. J. Brown, H. Budin, Multiple positive solutions for a class of nonlinear boundary value problems, J. Math. Anal. Appl., 60 (1977), 329–338. https://doi.org/10.1016/0022-247X(77)90023-3 doi: 10.1016/0022-247X(77)90023-3
[6]	K. J. Brown, H. Budin, On the existence of positive solutions for a class of semilinear elliptic boundary value problems, SIAM J. Math. Anal., 10 (1979), 875–883. https://doi.org/10.1137/0510082 doi: 10.1137/0510082
[7]	S. Y. Cheng, S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. Math., 104 (1976), 407–419. https://doi.org/10.2307/1970963 doi: 10.2307/1970963
[8]	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. https://doi.org/10.1515/ans-2012-0310 doi: 10.1515/ans-2012-0310
[9]	C. Corsato, F. Obersnel, P. Omari, S. Rivetti, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl., 405 (2013), 227–239. https://doi.org/10.1016/j.jmaa.2013.04.003 doi: 10.1016/j.jmaa.2013.04.003
[10]	G. W. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differ. Equ., 55 (2016), 1–17. https://doi.org/10.1007/s00526-016-1012-9 doi: 10.1007/s00526-016-1012-9
[11]	H. L. Gao, J. Xu, Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation, Bound. Value Probl., 81 (2021), 1–10. https://doi.org/10.1186/s13661-021-01558-x doi: 10.1186/s13661-021-01558-x
[12]	Z. Q. He, L. Y. Miao, Uniqueness and multiplicity of positive solutions for one-dimensional prescribed mean curvature equation in Minkowski space, AIMS Math., 5 (2020), 3840–3850. https://doi.org/10.3934/math.2020249 doi: 10.3934/math.2020249
[13]	Z. Q. He, L. Y. Miao, Existence and multiplicity of positive radial solutions for a Minkowski-curvature Dirichlet problem, Mediterr. J. Math., 19 (2022), 1–11. https://doi.org/10.1007/s00009-022-02040-3 doi: 10.1007/s00009-022-02040-3
[14]	Z. Q. He, L. Y. Miao, $S$-shaped connected component of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weight, Bull. Iran. Math. Soc., 48 (2022), 213–225. https://doi.org/10.1007/s41980-020-00512-4 doi: 10.1007/s41980-020-00512-4
[15]	S. Y. Huang, Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications, J. Differ. Equ., 264 (2018), 5977–6011. https://doi.org/10.1016/j.jde.2018.01.021 doi: 10.1016/j.jde.2018.01.021
[16]	S. Y. Huang, Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application, Commun. Pure Appl. Anal., 17 (2018), 1271–1294. https://doi.org/10.3934/cpaa.2018061 doi: 10.3934/cpaa.2018061
[17]	S. Y. Huang, Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications, Discrete Cont. Dyn. Syst., 39 (2019), 3443–3462. https://doi.org/10.3934/dcds.2019142 doi: 10.3934/dcds.2019142
[18]	S. Y. Huang, M. S. Hwang, Bifurcation curves of positive solutions for the Minkowski-curvature problem with cubic nonlinearity, Electron. J. Qual. Theory Differ. Equ., 41 (2021), 1–29. https://doi.org/10.14232/ejqtde.2021.1.41 doi: 10.14232/ejqtde.2021.1.41
[19]	T. Laetsch, J. B. Keller, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13. https://doi.org/10.1512/iumj.1970.20.20001 doi: 10.1512/iumj.1970.20.20001
[20]	R. Y. Ma, Y. Q. Lu, Multiplicity of positive solutions for second order nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator, Adv. Nonlinear Stud., 15 (2015), 789–803. https://doi.org/10.1515/ans-2015-0403 doi: 10.1515/ans-2015-0403
[21]	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. https://doi.org/10.1016/j.jfa.2016.01.020 doi: 10.1016/j.jfa.2016.01.020
[22]	R. Y. Ma, M. Xu, $S$-shaped connected component for a nonlinear Dirichlet problem involving mean curvature operator in one-dimension Minkowski space, Bull. Korean Math. Soc., 55 (2018), 1891–1908. https://doi.org/10.4134/BKMS.b180011 doi: 10.4134/BKMS.b180011
[23]	A. E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39–56. https://doi.org/10.1007/BF01404755 doi: 10.1007/BF01404755
[24]	X. M. Zhang, M. Q. Feng, Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space, Commun. Contemp. Math., 21 (2019), 1–17. https://doi.org/10.1142/S0219199718500037 doi: 10.1142/S0219199718500037

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