Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T17:47:41.840Z Has data issue: false hasContentIssue false
  • English
  • Français

OSCILLATION CRITERIA FOR CERTAIN DAMPED PDE'S WITH p-LAPLACIAN

Published online by Cambridge University Press:  01 January 2008

ZHITING XU
ZHITING XU*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R. China e-mail: xztxhyyj@pub.guangzhou.gd.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Some oscillation criteria are obtained for the damped PDE with p-Laplacian The results established here are extensions of some classical oscillation theorems due to Fite-Wintner and Kamenev for second order ordinary differential equations, and improve and complement recent results of Mařík and Usami.

Keywords

35J6034C3534K25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 1 , January 2008 , pp. 129 - 142
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Agarwal, R. P., Grace, S. R. and O'Regan, D., Oscillation theory for difference and functional differential equations (Kluwer, 2000).CrossRefGoogle Scholar
2.Agarwal, R. P., Grace, S. R. and O'Regan, D., Oscillation theory for second order linear, half-linear, superlinaer and sublinear dynamic equations (Kluwer, 2002).CrossRefGoogle Scholar
3.Díaz, J. I., Nonlinear partial differential equations and free boundaries, Vol. I. Elliptic equations, Pitman, London, 1985.Google Scholar
4.Došlý, O. and Mařík, R., Nonexistence of positive solutions of PDE's with p-Laplacian, Acta. Math. Hungar. 90 (2001), 89–107.CrossRefGoogle Scholar
5.Fite, W. B., Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc. 19 (1918), 341–352.CrossRefGoogle Scholar
6.Hardy, G., Littlewood, J. E. and Pólya, G., Inequalties, Second edition (Cambridge University Press, 1999).Google Scholar
7.Hartman, P., Ordinary differential equations (Wiley, 1964).Google Scholar
8.Kusano, T., Jaroš, J. and 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. 40 (2007), 381–395.CrossRefGoogle Scholar
9.Ladde, G. S., Lakshmikantham, V. and Zhang, B. G., Oscillation theory of differential equations with deviating argument (Marcel Dekker, 1987).Google Scholar
10.Kamenev, I. V., Oscillation of solutions of a second order differential equation with an “integrally small’ coefficient, Differencial'nye Uravnenija. 13 (1977), 2141–2148 (in Russian).Google Scholar
11.Mařík, R., Oscillation criteria for PDE with p-Laplacian via the Riccati technique, J. Math. Anal. Appl. 248 (2000), 290–308.CrossRefGoogle Scholar
12.Mařík, R., Hartman-Wintner type theorem for PDE with p-Laplacian, Proc. Colloq. Qual. Theory Differ. Equ. 18 (2000), 1–7.Google Scholar
13.Mařík, R., Riccati-type inequality and oscillation criteria for a half-linear PDE with damping, Electron J. Diff. Eqs. 11 (2004), 1–17.Google Scholar
14.Mařík, R., Integral averages and oscillation criteria for a half-linear partial differential equation, Appl. Math. Comput. 150 (2004), 69–87.Google Scholar
15.Mařík, R., Interval-type oscillation criteria for half-linear PDE with damping, Appl. Appl. Math. 1 (2006), 1–10.Google Scholar
16.Noussair, E. S. and Swanson, C. A., Oscillation of semilinear elliptic inequalities by Riccati transformation, Canad. J. Math. 32 (4) (1980) 908–923.CrossRefGoogle Scholar
17.Swanson, C. A., Comparison and oscillatory theory of linear differential equations (Academic Press, 1968).Google Scholar
18.Swanson, C. A., Semilinear second order elliptic oscillation, Canad. Math. Bull. 22 (1979), 139–157.CrossRefGoogle Scholar
19.Usami, H., Some oscillation theorems for a class of quasilinear elliptic equations, Ann. Math. Pura. Appl. 175 (1998) 277–283.CrossRefGoogle Scholar
20.Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115–117.CrossRefGoogle Scholar
21.Xu, Z., Oscillation of second order elliptic partial differential equations with a “weakly integrally small” coefficient, J. Sys & Math. Scis. 18 (1998), 478–484. (in Chinese).Google Scholar
22.Xu, Z., Oscillation properties for quasilinear elliptic equations in divergence form, J. Sys & Math. Scis. 24 (2004), 85–95 (in Chinese).Google Scholar
23.Xu, Z., Riccati inequality and oscillation criteria for PDE with p-Laplacian, J. Inequal Appl. 2006, Art. ID 63061, 1–10.Google Scholar
24.Xu, Z. and Xing, H., Oscillation criteria of Kamenev-type for PDE with p-Laplacian, Appl. Math. Comput. 145 (2003), 735–745.Google Scholar
25.Xu, Z. and Xing, H., Oscillation criteria for PDE with p-Laplacian involving general means, Ann. Mat. Pura Appl. 184 (2005), 395–406.CrossRefGoogle Scholar
26.Zhang, B. G., Zhao, T. and Lalli, B. S., Oscillation criteria for nonlinear second order elliptic differential equations, Chin. Ann. Math. Ser. B. 17 (1996), 89–102.Google Scholar