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S-Shaped Connected Component of Positive Solutions for a Minkowski-Curvature Dirichlet Problem with Indefinite Weight

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Abstract

In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weight. By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to the parameter \(\lambda \). In particular, we obtain the existence of at least three positive solutions for \(\lambda \) being in a certain interval.

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References

  1. Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87, 131–152 (1982–1983)

  2. Bereanu, C., Jebelean, P., Torres, P.J.: Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J. Funct. Anal. 264(1), 270–287 (2013)

    Article  MathSciNet  Google Scholar 

  3. Boscaggin, A., Garrione, M.: Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball. Commun. Contemp. Math. 21(2), 1850006, 18 (2019)

  4. Boscaggin, A., Colasuonno, F., Noris, B.: Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete Contin. Dyn. Syst. Ser. S 13(7), 1921–1933 (2020)

    MathSciNet  MATH  Google Scholar 

  5. Boscaggin, A., Feltrin, G., Zanolin, F.: Positive solutions for a Minkowski-curvature equation with indefinite weight and super-exponential nonlinearity (2020). arXiv:2007.00338

  6. Boscaggin, A., Feltrin, G.: Positive periodic solutions to an indefinite Minkowski-curvature equation (2020). arXiv:1805.06659

  7. Boscaggin, A., Feltrin, G.: Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefinite weight. Nonlinear Anal. 196, 111807 (2020)

    Article  MathSciNet  Google Scholar 

  8. Cao, X.F., Dai, G.W., Zhang, N.: Global structure of positive solutions for problem with mean curvature operator on an annular domain. Rocky Mt. J. Math. 48, 1799–1814 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Coelho, I., Corsato, C., Obersnel, F., Omari, P.: Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation. Adv. Nonlinear Stud. 12(3), 621–638 (2012)

    Article  MathSciNet  Google Scholar 

  10. Coelho, I., Corsato, C., Rivetti, S.: Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball. Topol. Methods Nonlinear Anal. 44(1), 23–39 (2014)

    Article  MathSciNet  Google Scholar 

  11. Corsato, C., Obersnel, F., Omari, P.: The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space. Georg. Math. J. 24(1), 113–134 (2017)

    Article  MathSciNet  Google Scholar 

  12. Dai, G.W., Wang, J.: Nodal solutions to problem with mean curvature operator in Minkowski space. Differ. Integral Equations 30, 463–480 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Gaudenzi, M., Habets, P., Zanolin, F.: Positive solutions of superlinear boundary value problems with singular indefinite weight. Commun. Pure Appl. Anal. 2, 411–423 (2003)

    Article  MathSciNet  Google Scholar 

  14. He, Z.Q., Ma, R.Y., Xu, M.: Three positive solutions for second-order periodic boundary value problems with sign-changing weight. Bound. Value Probl. 2018, 93 (2018)

    Article  MathSciNet  Google Scholar 

  15. Huang, S.-Y.: Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications. J. Differ. Equations 264, 5977–6011 (2018)

    Article  MathSciNet  Google Scholar 

  16. Li, H., Yeh, C.: Sturmian comparison theorem for half-linear second-order differential equations. Proc. R. Soc. Edinbu. Sect. A 125, 1193–1204 (1995)

    Article  MathSciNet  Google Scholar 

  17. Ma, R.Y., Wei, L.P., Chen, Z.C.: Evolution of bifurcation curves for one-dimensional Minkowski-curvature problem. App. Math. Lett. 103, 106176 (2020)

    Article  MathSciNet  Google Scholar 

  18. Ma, R.Y., Gao, H.L., Lu, Y.Q.: Global structure of radial positive solutions for a prescribed mean curvature problem in a ball. J. Funct. Anal. 270(7), 2430–2455 (2016)

    Article  MathSciNet  Google Scholar 

  19. Ma, R.Y., Xu, M.: \(S\)-shaped connected component for a nonlinear Dirichlet problem involving mean curvature operator in one-dimension Minkowski space. Bull. Korean Math. Soc. 55, 1891–1908 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Sim, I., Tanaka, S.: Three positive solutions for one-dimensional \(p\)-Laplacian problem with sign-changing weight. Appl. Math. Lett. 49, 42–50 (2015)

    Article  MathSciNet  Google Scholar 

  21. Xu, M., Ma, R.Y.: \(S\)-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain. Open Math. 17, 929–941 (2019)

    Article  MathSciNet  Google Scholar 

  22. Zhang, X.M., Feng, M.Q.: Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space. Commun. Contemp. Math. 21(3), 1850003, 17 (2019)

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Authors and Affiliations

  1. Department of Basic Teaching and Research, Qinghai University, Xining, 810016, People’s Republic of China

    Zhiqian He

  2. College of Mathematics and Statistics, Qinghai Nationalities University, Xining, 810007, People’s Republic of China

    Liangying Miao

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  1. Zhiqian He
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  2. Liangying Miao
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Correspondence to Zhiqian He.

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Communicated by Majid Gazor.

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Z. He: Supported by the NSFC (No. 11861056).

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He, Z., Miao, L. S-Shaped Connected Component of Positive Solutions for a Minkowski-Curvature Dirichlet Problem with Indefinite Weight. Bull. Iran. Math. Soc. 48, 213–225 (2022). https://doi.org/10.1007/s41980-020-00512-4

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  • DOI: https://doi.org/10.1007/s41980-020-00512-4

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