Skip to main content
SpringerLink
Log in

Averaging technique for the oscillation of second order damped elliptic equations

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

By using the averaging technique, we establish some oscillation theorems for the second order damped elliptic differential equation

$$ \sum\limits_{i,j = 1}^N {D_i [a_{ij} (x)D_j y] + } \sum\limits_{i = 1}^N {b_i (x)D_j y + c(x)f(y) = 0,} $$

which extend and improve some known results in the literature.

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. Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983

    MATH  Google Scholar 

  2. Fite, W. B.: Concerning the zeros of the solutions of certain differential equations. Trans. Amer. Math. Soc., 19, 341–352 (1918)

    Article  MathSciNet  Google Scholar 

  3. Wintner, A.: A criterion of oscillatory stablity. Quart. Appl. Math., 7, 115–117 (1949)

    MATH  MathSciNet  Google Scholar 

  4. Kamenev, I. V.: An integral criterion for oscillation of linear differential equations. Mat. Zametki., 23, 249–251 (1978) (in Russian)

    MATH  MathSciNet  Google Scholar 

  5. Yan, J.: Oscillation theorems for second order linear differential equations with damping. Proc. Amer. Math. Soc., 98, 276–282 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, B. S.: Oscillation criteria for nonlinear second order damping differential equations. Acta Math. Appl. Sinica., 15, 166–173 (1992) (in Chinese)

    MATH  MathSciNet  Google Scholar 

  7. Philos, Ch. G.: Oscillation theorems for linear differential equations of second order. Arch. Math., (Basel), 53, 482–492 (1989)

    MATH  MathSciNet  Google Scholar 

  8. Wong, J. S. W.: On Kamenev-tye oscillation theorems for second order differential equations with damping. J. Math. Anal. Appl., 258, 244–257 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kong, Q.: Interval criteria for oscillation of second order linear differential equations. J. Math. Anal. Appl., 229, 258–270 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Li, W. T., Agarwal, R. P.: Interval criterion for oscillation of second order nonlinear differential equations with damping. Comput. Math. Appl., 40, 217–230 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Noussair, E. S., Swanson, C. A.: Oscillation of semilinear elliptic inequalities by Riccati transformation. Canad. J. Math., 32(4), 908–923 (1980)

    MATH  MathSciNet  Google Scholar 

  12. Swanson, C. A.: Semilinear second order elliptic oscillation. Canad. Math. Bull., 22(2), 139–157 (1979)

    MATH  MathSciNet  Google Scholar 

  13. Xu, Z. T.: Oscillation of second order elliptic partial differential equations with a ‘weakly integrally small’ coefficient. J. Sys. Math. Scis., 18(4), 478–484 (1998) (in Chinese)

    MATH  Google Scholar 

  14. Xu, Z. T.: Riccati techniques and oscillation of semilinear elliptic equations. Chin. J. Contemp. Math., 24(4), 329–340 (2003)

    Google Scholar 

  15. Xu, Z. T., Ma, D. K., Jia, B. G.: Oscillation theorems for elliptic equations of second order. Acta Mathematica Scientia (A), (in Chinese), 24(2), 144–151 (2004)

    MathSciNet  Google Scholar 

  16. Zhang, B. G., Zhao, T., Lalli, B. S.: Oscillation criteria for nonlinear second order elliptic differential equations. Chin Ann of Math (B)., 1(17), 89–101 (1996)

    MathSciNet  Google Scholar 

  17. Zhuang, R. K., Wang, Q. R., Zhu, S. M.: Domain oscillation criteria for second order elliptic differential equations. Acta Mathematic Sinica, Chinese Series, 48(3), 527–534 (2005)

    MATH  MathSciNet  Google Scholar 

  18. Xu, Z. T., Jia, B. G., Ma, D. K.: Oscillation theorems for elliptic equations with damping. Appl. Math. Comput., 156, 93–106 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Xu, Z. T.: Oscillation and nonoscillations of solutions of secon order damped elliptic differential equations. Acta Mathematica Sincia, Chines Series, 47(6), 1099–1106 (2004)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R. China

    Zhi Ting Xu

Authors
  1. Zhi Ting Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi Ting Xu.

Additional information

Supported partly by the NFS of China (10571064) and the NSF of Guangdong Province (04010364)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, Z.T. Averaging technique for the oscillation of second order damped elliptic equations. Acta. Math. Sin.-English Ser. 24, 829–842 (2008). https://doi.org/10.1007/s10114-007-5397-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-007-5397-4

Keywords

MR(2000) Subject Classification

Search

Navigation