Skip to main content
SpringerLink
Log in

Gelfand problem on a large spherical cap

  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

Abstract

We study the behaviour of the minimal solution to the Gelfand problem on a spherical cap under the Dirichlet boundary conditions. The asymptotic behaviour of the solution is discussed as the cap approaches the whole sphere. The results are based on the sharp estimate of the torsion function of the spherical cap in terms of the principle eigenvalue which we derive in this work.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Agmon, S.: On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. In: Methods of functional analysis and theory of elliptic equations (Naples, 1982), Liguori, Naples, pp. 19–52 (1983)

  2. Bandle, C.: Existence theorems, qualitative results and a priori bounds for a class of nonlinear Dirichlet problems. Arch. Ration. Mech. Anal. 58, 219–238 (1975)

    Article  MathSciNet  Google Scholar 

  3. Bandle, C., Benguria, R.: The Brézis–Nirenberg problem on \(\mathbb{S}^3\). J. Differ. Equ. 178, 264–279 (2002)

    Article  Google Scholar 

  4. Bandle, C., Kabeya, Y., Ninomiya, H.: Imperfect bifurcations in nonlinear elliptic equations on spherical caps. Commun. Pure Appl. Anal. 10, 1189–1208 (2010)

    Article  MathSciNet  Google Scholar 

  5. Beals, R., Wong, R.: Special Functions, A Graduate Text, Cambridge Studies in Advanced Mathematics, vol. 126. Cambridge (2010)

  6. Brezis, H., Cazenave, T., Martel, Y., Ramiandrisoa, A.: Blow up for \(u_t-\Delta u=g(u)\) revisited. Adv. Differ. Equ. 1, 73–90 (1996)

    MATH  Google Scholar 

  7. Crandall, M.G., Rabinowitz, P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58, 207–218 (1975)

    Article  MathSciNet  Google Scholar 

  8. Dupaigne, L.: Stable Solutions of Elliptic Partial Differential Equations, p. xiv+321. Chapman & Hall/CRC, Boca Raton (2011)

    Book  Google Scholar 

  9. Frank-Kamenetskii, D.A.: Diffusion and Heat Transfer in Chemical Kinetics. Princeton University Press, Princeton (1955)

    Book  Google Scholar 

  10. Gelfand, I.M.: Some problems in the theory of quasi-linear equations. Am. Math. Soc. Transl. 2(29), 295–381 (1963)

    Google Scholar 

  11. Joseph, D., Lundgren, T.: Quasilinear Dirichlet problem driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1972/73)

  12. Kabeya, Y., Kawakami, T., Kosaka, A., Ninomiya, H.: Eigenvalues of the Laplace–Beltrami operator on a large spherical cap under the Robin problem. Kodai Math. J. 37, 620–645 (2014)

    Article  MathSciNet  Google Scholar 

  13. Keller, H.B., Cohen, D.S.: Some positone problems suggested by nonlinear heat generation. J. Math. Mech. 16, 1361–1376 (1967)

    MathSciNet  MATH  Google Scholar 

  14. Keener, J.P., Keller, H.B.: Positive solutions of convex nonlinear eigenvalue problems. J. Differ. Equ. 16, 103–125 (1974)

    Article  MathSciNet  Google Scholar 

  15. Kosaka, A.: Bifurcations of solutions to semilinear elliptic problems on \({ S}^2\) with a small hole. Sci. Math. Jpn. 78, 17–42 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Liskevich, V., Lyakhova, S., Moroz, V.: Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains. Adv. Differ. Equ. 11, 361–398 (2006)

    MathSciNet  MATH  Google Scholar 

  17. MacDonald, H.M.: Zeros of the spherical harmonics \(P_n^{m}(\mu )\) considered as a function of \(n\). Proc. Lond. Math. Soc. 31, 264–278 (1899)

    Article  MathSciNet  Google Scholar 

  18. Moriguchi, S., Udagawa, K., Hitotsumatsu, S.: Mathematical Formula III. Iwanami Shoten, Tokyo (1960). (in Japanese)

    Google Scholar 

  19. Sattinger, D. H.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21, 979–1000 (1971/72)

Download references

Acknowledgements

The authors are grateful to the anonymous referee for their comments which helped to improve the exposition in the paper. Part of this research was carried out while VM was visiting Osaka Prefecture University. VM thanks the Department of Mathematical Sciences for its support and hospitality. YK is supported in part by JSPS KAKENHI Grant Number 19K03588.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Osaka Prefecture University, Gakuencho, Sakai, 599-8531, Japan

    Yoshitsugu Kabeya

  2. Department of Mathematics, Swansea University, Fabian Way, Swansea, SA1 8EN, Wales, UK

    Vitaly Moroz

Authors
  1. Yoshitsugu Kabeya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vitaly Moroz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vitaly Moroz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kabeya, Y., Moroz, V. Gelfand problem on a large spherical cap. J Elliptic Parabol Equ 7, 1–23 (2021). https://doi.org/10.1007/s41808-020-00091-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41808-020-00091-9

Keywords

Mathematics Subject Classification

Search

Navigation