Skip to main content
SpringerLink
Log in

The nonexistence of positive solutions to some elliptic equations

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The main result of the paper is the nonexistence of integer positive solutions in wide classes of quasilinear elliptic equations whose model examples are equations of the form {

$$\sum\limits_{i = 1}^n {\frac{\partial }{{\partial _x }}\left( {\left| {\nabla _u } \right|^{a - 2} u_{x_i } } \right) = - \left| u \right|^{q - 1} u} $$

}, where α>1 andq are some fixed real numbers.

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. B. Gidas and J. Spruck, “Global and local behavior of positive solutions of nonlinear elliptic equations,”Comm. Pure Appl. Math.,34, 525–598 (1981).

    Article  MATH  Google Scholar 

  2. W.-M. Ni and J. Serrin, “Nonexistence theorems for quasilinear partial differential equations,”Rend. Circ. Mat. Palermo,5, No. 2, 171–185 (1985).

    Google Scholar 

  3. M. F. Bidaut-Veron, “Global and local behavior of solutions of quasilinear equations of Emden-Fowler type,”Arch. Rational Mech. Anal.,107, 293–324 (1989).

    Article  MATH  Google Scholar 

  4. Ph. Clement, R. Monasevich, and E. Mitidieri, “Positive solutions of quasilinear systems via blow-up,”Comm. Partial Differential Equations,18, 2071–2106 (1993).

    Article  MATH  Google Scholar 

  5. H. Berestycki, Dolcetta J. Capuzzo, and L. Nirenberg, “Superlinear indefinite elliptic problems and nonlinear Liouville theorems,”Topol. Methods Nonlinear Anal.,4, 59–78 (1995).

    Google Scholar 

  6. E. MitidieriDifferential Integral Equations,9, 465–479 (1996).

    MATH  Google Scholar 

  7. V. V. Kurta,Some Problems of the Qualitative Theory of Nonlinear Second-Order Differential Equations [in Russian], Doctorate thesis in the physico-mathematical sciences, Moscow (1994).

  8. V. V. Kurta,Some Problems of the Qualitative Theory of Nonlinear Second-Order Differential Equations [in Russian], Summary of the doctorate thesis in the physico-mathematical sciences, Moscow (1995).

  9. V. V. Kurta, “Nonexistence of integer positive solutions of elliptic equations,”Uspekhi Mat. Nauk [Russian Math. Surveys],50, No. 4, 127 (1995).

    Google Scholar 

  10. V. M. Miklyukov, “A new approach to the Bernstein theorem and related problems for equations of minimal surface type,”Mat. Sb. [Math. USSR-Sb.],108 (150), No. 2, 268–289 (1979).

    Google Scholar 

  11. V. M. Miklyukov, “Capacitance and the generalized maximum principle for elliptic-type quasilinear equations,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],250, No. 6, 1318–1320 (1980).

    Google Scholar 

  12. V. M. Gol'dshtein and Yu. G. Reshetnyak,Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  13. A. N. Kolmogorov and S. V. Fomin,Elements of Function Theory and Functional Analysis [in Russian], Nauka, Moscow (1989).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics and Mechanics, National Academy of Sciences of the Ukraine, Donetsk

    V. V. Kurta

Authors
  1. V. V. Kurta
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 552–561, April, 1999.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kurta, V.V. The nonexistence of positive solutions to some elliptic equations. Math Notes 65, 462–469 (1999). https://doi.org/10.1007/BF02675360

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02675360

Key words

Search

Navigation