Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Variation of the instability index on stationary curves of elliptic equations depending on a parameter

  • 16 Accesses

Abstract

The set of solutions of the equation A(u, γ)=0 in the case of general position consists of smooth curves. Solutions of a quasilinear elliptic equation with large instability index are constructed and an estimate obtained for the number of turning points of the solution curves.

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

Literature cited

  1. 1.

    D. H. Sattinger, “Stability of bifurcating solutions by Leray-Schauder degree,” Arc. Rat. Mech. Anal.,43, No. 2, 154–166 (1971).

  2. 2.

    M. G. Crandall and P. H. Rabinowitz, “Bifurcation perturbation of simple eigenvalues, and linearized stability,” Arch. Rat. Mech. Anal.,52, No. 2, 161–180 (1973).

  3. 3.

    S. Smale, “An infinite dimensional version of Sard's theorem,” Am. J. Math.,87, No. 4, 861–866 (1965).

  4. 4.

    A. V. Babin and M. I. Visik, “Regular attractors of semigroups and evolution equations,” J. Math. Pures Appl.,62, No. 4, 441–491 (1983).

  5. 5.

    M. A. Shubin, Pseudodifferential Operators and Spectral Theory [in Russian], Nauka, Moscow (1978).

  6. 6.

    A. V. Babin and M. I. Vishik, “Attractors of partial differential evolution equations and bounds for their dimensions,” Usp. Mat. Nauk,38, No. 4, 133–187 (1983).

  7. 7.

    I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators [in Russian], Nauka, Moscow (1965).

Download references

Additional information

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 47–58, 1987.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Babin, A.V., Vishik, M.I. Variation of the instability index on stationary curves of elliptic equations depending on a parameter. J Math Sci 47, 2516–2525 (1989). https://doi.org/10.1007/BF01102995

Download citation

Keywords

  • Elliptic Equation
  • General Position
  • Smooth Curf
  • Solution Curve
  • Quasilinear Elliptic Equation