Skip to main content
SpringerLink
Log in

Bifurcation problems for equations of elliptic type with discontinuous nonlinearities

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We consider the problem of the existence of semiregular solutions to the main boundary-value problems for second-order equations of elliptic type with a spectral parameter and discontinuous nonlinearities. A variational method is used to obtain the theorem on the existence of solutions and properties of the “separating” set for the problems under consideration. The results obtained are applied to the Goldshtik problem.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. A. Krasnosel’skii and A.V. Pokrovskii, “Regular solutions of equations with discontinuous nonlinearities,” Dokl. Akad. Nauk SSSR 226(3), 506–509 (1976) [SovietMath. Dokl. 17, 128–132 (1976)].

    MathSciNet  Google Scholar 

  2. V. N. Pavlenko, Equations and Variational Inequalities with Discontinuous Nonlinearities, Doctoral Dissertation in Mathematics and Physics (Chelyabinsk, 1995) [in Russian].

  3. D. K. Potapov, “On a “separating” set for higher-order equations of elliptic type with discontinuous nonlinearities,” Differ. Uravn. 46(3), 451–453 (2010) [Differ. Equations 46 (3), 458–460 (2010)].

    MathSciNet  Google Scholar 

  4. M. A. Gol’dshtik, “A mathematical model of separated flows in an incompressible liquid,” Dokl. Akad. Nauk SSSR 147(6), 1310–1313 (1962) [Soviet Phys. Dokl. 7, 1090–1093 (1963)].

    Google Scholar 

  5. V. N. Pavlenko and D. K. Potapov, “Existence of a ray of eigenvalues for equations with discontinuous operators,” Sibirsk. Mat. Zh. 42(4), 911–919 (2001) [SiberianMath. J. 42 (4), 766–773 (2001)].

    MathSciNet  Google Scholar 

  6. D. K. Potapov, Problems with a Spectral Parameter and a Discontinuous Nonlinearity (IBP, St. Petersburg, 2008) [in Russian].

    Google Scholar 

  7. D. K. Potapov, “On an upper bound for the value of the bifurcation parameter in eigenvalue problems for elliptic equations with discontinuous nonlinearities,” Differ.Uravn. 44(5), 715–716 (2008) [Differ. Equations 44 (5), 737–739 (2008)].

    MathSciNet  Google Scholar 

  8. S. A. Marano, “Elliptic eigenvalue problems with highly discontinuous nonlinearities,” Proc. Amer. Math. Soc. 125(10), 2953–2961 (1997).

    Article  MathSciNet  Google Scholar 

  9. D. K. Potapov, “A mathematical model of separated flows in an incompressible fluid,” Izv. RAEN Ser. MMMIU 8(3–4), 163–170 (2004).

    Google Scholar 

  10. D. K. Potapov, “Continuous approximations of Gol’dshtik’s model,” Mat. Zametki 87(2), 262–266 (2010) [Math. Notes 87 (2), 244–247 (2010)].

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    D. K. Potapov

Authors
  1. D. K. Potapov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. K. Potapov.

Additional information

Original Russian Text © D. K. Potapov, 2011, published in Matematicheskie Zametki, 2011, Vol. 90, No. 2, pp. 280–284.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Potapov, D.K. Bifurcation problems for equations of elliptic type with discontinuous nonlinearities. Math Notes 90, 260 (2011). https://doi.org/10.1134/S000143461107025X

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1134/S000143461107025X

Keywords

Search

Navigation