Abstract
In this paper, we study the number of classical positive radial solutions for Dirichlet problems of type
where λ is a positive parameter,
-
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: Supported by the Natural Science Foundation of Qinghai Province (2021-ZJ-957Q), the National Natural Science Foundation of China (No. 11671322).
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] R. Bartnik and L. Simon, “Spacelike hypersurfaces with prescribed boundary values and mean curvature,” Commun. Math. Phys., vol. 87, pp. 131–152, 1982–1983. https://doi.org/10.1007/bf01211061.Search in Google Scholar
[2] C. Bereanu, P. Jebelean, and P. J. Torres, “Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space,” J. Funct. Anal., vol. 264, no. 1, pp. 270–287, 2013. https://doi.org/10.1016/j.jfa.2012.10.010.Search in Google Scholar
[3] C. Bereanu, P. Jebelean, and J. Mawhin, “Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces,” Proc. Am. Math. Soc., vol. 137, pp. 171–178, 2009.10.1090/S0002-9939-08-09612-3Search in Google Scholar
[4] C. Bereanu, P. Jebelean, and J. Mawhin, “The Dirichlet problem with mean curvature operator in Minkowski space-a variational approach,” Adv. Nonlinear Stud., vol. 14, pp. 315–326, 2014. https://doi.org/10.1515/ans-2014-0204.Search in Google Scholar
[5] C. Bereanu, P. Jebelean, and P. J. Torres, “Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space,” J. Funct. Anal., vol. 265, pp. 644–659, 2013. https://doi.org/10.1016/j.jfa.2013.04.006.Search in Google Scholar
[6] I. Coelho, C. Corsato, and S. Rivetti, “Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball,” Topol. Methods Nonlinear Anal., vol. 44, no. 1, pp. 23–39, 2014.10.12775/TMNA.2014.034Search in Google Scholar
[7] G. W. Dai, “Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space,” Calc. Var. Partial Differ. Equ., vol. 55, no. 4, pp. 1–17, 2016. https://doi.org/10.1007/s00526-016-1012-9.Search in Google Scholar
[8] B. H. Feng, R. P. Chen, and Q. X. Wang, “Instability of standing waves for the nonlinear Schrödinger-Poisson equation in the L2-critical case,” J. Dynam. Differ. Equ., vol. 32, no. 3, pp. 1357–1370, 2020. https://doi.org/10.1007/s10884-019-09779-6.Search in Google Scholar
[9] Z. T. Liang, L. Duan, and D. D. Ren, “Multiplicity of positive radial solutions of singular Minkowski-curvature equations,” Arch. Math., vol. 113, no. 4, pp. 415–422, 2019. https://doi.org/10.1007/s00013-019-01341-6.Search in Google Scholar
[10] Z. T. Liang and Y. J. Yang, “Radial convex solutions of a singular dirichlet problem with the mean curvature operator in Minkowski space,” Acta Math. Sci. Ser. B (Engl. Ed.), vol. 39, no. 2, pp. 395–402, 2019. https://doi.org/10.1007/s10473-019-0205-7.Search in Google Scholar
[11] R. Y. Ma, H. L. Gao, and Y. Q. Lu, “Global structure of radial positive solutions for a prescribed mean curvature problem in a ball,” J. Funct. Anal., vol. 270, no. 7, pp. 2430–2455, 2016. https://doi.org/10.1016/j.jfa.2016.01.020.Search in Google Scholar
[12] M. H. Pei and L. B. Wang, “Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space,” Appl. Math. Lett., vol. 60, pp. 50–55, 2016. https://doi.org/10.1016/j.aml.2016.04.001.Search in Google Scholar
[13] M. H. Pei and L. B. Wang, “Positive radial solutions of a mean curvature equation in Minkowski space with strong singularity,” Proc. Am. Math. Soc., vol. 145, no. 10, pp. 4423–4430, 2017. https://doi.org/10.1090/proc/13587.Search in Google Scholar
[14] M. H. Pei, L. B. Wang, and X. Z. Lv, Existence and multiplicity of positive solutions of a one-dimensional mean curvature equation in Minkowski space. Bound. Value Probl., pp. 43, 2018. https://doi.org/10.1186/s13661-018-0963-5.Search in Google Scholar
[15] X. M. Zhang and M. Q. Feng, “Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space,” Commun. Contemp. Math., vol. 21, pp. 1850003, 2019. https://doi.org/10.1142/s0219199718500037.Search in Google Scholar
[16] S.-Y. Huang, “Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications,” J. Differ. Equ., vol. 264, pp. 5977–6011, 2018. https://doi.org/10.1016/j.jde.2018.01.021.Search in Google Scholar
[17] K. Deimling, Nonlinear Functional Analysis, Berlin, Springer, 1985.10.1007/978-3-662-00547-7Search in Google Scholar
[18] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Orlando, FL, Academic Press, 1988.Search in Google Scholar
[19] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Groningen, P. Noordhofi Ltd, 1964.Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston