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Oscillation criteria for second order mixed nonlinear elliptic equations

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Abstract

By using the refinement of the standard integral averaging technique, we obtain some oscillation criteria for second order mixed nonlinear elliptic equations. The results established in this paper extend and improve some existing oscillation criteria for half-linear PDE in the literature.

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References

  1. Došlý, O., Mařík, R. Nonexistence of positive solutions of PDE’s with p-Laplacian. Acta. Math. Hungar., 90(1-2): 89–107 (2001)

    MATH  MathSciNet  Google Scholar 

  2. Gilbarg, D., Trudinger, N.S. Elliptic Partial Differential Equations of Second order. Springer-Verlag, Herdelberg, New York, 1971

    Google Scholar 

  3. Hardy, G., Littewood, J.E., Pólya, G. Inequalities, Second Edition. Cambridge Mathematical Library, Cambridge, 1988

    Google Scholar 

  4. Jaroš, J., Kusano, T., Yoshida, N. Picone-type identity for nonlinear elliptic equations and their applications. J. Inequal. Appl., 6: 387–404 (2001)

    MATH  MathSciNet  Google Scholar 

  5. Jaroš, J., Kusano, T., Yoshida, N. Picone-type identity for half-linear elliptic equations and their applications. Adv. Math. Sci. Appl., 12: 709–724 (2002)

    MATH  MathSciNet  Google Scholar 

  6. Kusano, T., Jaroš, J., Yoshida, N. A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order. Nonlinear Anal. TMA., 40: 381–395 (2003)

    Article  Google Scholar 

  7. Mařík, R. Oscillation criteria for PDE with p-Laplacian via the Riccati technique. J. Math. Anal. Appl., 248: 290–308 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mařík, R. Hartman-Wintner type theorem for PDE with p-Laplacian. EJQTDE. Proc. 6th Coll. QTDE, 18: 1–7 (2000)

    Google Scholar 

  9. Mařík, R. Integral averages and oscillation criteria for half-linear partial differential equation. Appl. Math. Comput., 150: 69–87 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mařík, R. Riccati-type inequality and oscillation criteria for a half-linear PDE with damping. Elect. J. Diff. Eqs., 11: 1–17 (2004)

    Google Scholar 

  11. Mařík, R. Ordinary differential equations in the oscillation theory of partial half-linear differential equations. J. Math. Anal. Appl., 338: 194–208 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mařík, R. Oscillation Theory of Partial Differential Equations with p-Laplacian. Folia Univ. Agric. et Silvic. Mendel. Brun. Brno., 2008

    Google Scholar 

  13. Mařík, R. Conjugacy crieria for half-linear ODE in the theorey of PDE with generalized p-Lapacian and mixed powers. Nonlinear Analysis., 73: 500–509 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Rogovchenko, Yu. V., Tuncay, F. Oscillation criteria for second order nonlinear differential equations with damping. Nonlinear Anal. TMA., 69: 208–221 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Usami, H. Some oscillation theorems for a class of quasilinear elliptic equations. Ann. Math. Pura Appl., 175: 277–283 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Xu, Z. Oscillation criteria for certain damped PDE with p-Laplacian. Glasgow Math. J., 50: 129–142 (2008)

    Article  MATH  Google Scholar 

  19. Xu, Z. Oscillation criteria for damped half-linear PDE via the integral operator. Math. Comput. Modelling., 48: 1227–1236 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  21. Xu, Z., Ma, D., Jia, B. Integral averaging technique for oscillation of elliptic equations of second order. J. Partial Diff. Eqs., 16: 193–203 (2003)

    MATH  MathSciNet  Google Scholar 

  22. Xu, Z., Xing, H. Oscillation criteria for PDE with p-Laplacian involving general means. Ann. Math. Pura Appl., 184(4): 395–406 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yoshida, N. Oscillation Theory of Partial Differential Equations. World Scientific Publishing Co. Pte. Ltd., 2008

    Book  MATH  Google Scholar 

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Correspondence to Zhi-ting Xu.

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Supported by the Doctoral Program of Higher Education of China (20094407110001).

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Xu, Zt. Oscillation criteria for second order mixed nonlinear elliptic equations. Acta Math. Appl. Sin. Engl. Ser. 31, 1049–1060 (2015). https://doi.org/10.1007/s10255-015-0526-9

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  • DOI: https://doi.org/10.1007/s10255-015-0526-9

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