Taiwanese Journal of Mathematics

OSCILLATION AND NONOSCILLATION OF SOLUTIONS OF PDE WITH $p$−LAPLACIAN

Zhiting Xu

Full-text: Open access

Abstract

Some necessary conditions are established for the nonoscillation of the following PDE with $p-$Laplacian \begin{equation*} \mbox{div}(\|\nabla y\|^{p-2}\nabla y)+c(x)|y|^{p-2}y=0. \end{equation*} Using these results, we obtain some oscillation criteria for the above equation.

Article information

Source
Taiwanese J. Math., Volume 13, Number 6B (2009), 2037-2049.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405656

Digital Object Identifier
doi:10.11650/twjm/1500405656

Mathematical Reviews number (MathSciNet)
MR2583536

Zentralblatt MATH identifier
1196.35095

Subjects
Primary: 35J60: Nonlinear elliptic equations 34C35 34K25: Asymptotic theory

Keywords
oscillation nonoscillation PDE with $p-$Laplacian Riccati transformation

Citation

Xu, Zhiting. OSCILLATION AND NONOSCILLATION OF SOLUTIONS OF PDE WITH $p$−LAPLACIAN. Taiwanese J. Math. 13 (2009), no. 6B, 2037--2049. doi:10.11650/twjm/1500405656. https://projecteuclid.org/euclid.twjm/1500405656


Export citation

References

  • [1.] J. I. D\'\tiny laz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. I. Elliptic Equations, Pitman, London, 1985.
  • [2.] O. Do\uslý and R. Mař\'\tiny lk, Nonexistence of positive solutions of PDE's with $p-$Laplacian, Acta. Math. Hungar., 90(1-2) (2001), 89-107.
  • [3.]G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd, Cambridge Univ Press, Cambridge, UK, 1999.
  • [4.]P. Hartman, Ordinary Differential Equations, New York, Wiley, 1982.
  • [5.]T. Kusano, J. Jaro\us and N. Yoshida, A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order, Nonl. Anal., 40 (2003), 381-395.
  • [6.]W. Leighton, The detection of the oscillation of solutions of a second order linear differential equations, Duck Math. J., 17 (1950), 57-61.
  • [7.]R. Mař\'\tiny lk, Oscillation criteria for PDE with $p-$Laplacian via the Riccati technique, J. Math. Anal. Appl., 248 (2000), 290-308.
  • [8.] R. Mař\'\tiny lk, Hartman-Wintner type theorem for PDE with $p-$Laplacian, EJQTDE. Proc. 6th Coll. QTDE., 18 (2000), 1-7.
  • [9.]R. Mař\'\tiny lk, Integral averagings and oscillation criteria for half-linear partial differential equation, Appl. Math. Comput., 150 (2004), 69-87.
  • [10.]E. S. Noussair and C. A. Swanson, Oscillation of semilinear elliptic inequalities by Riccati transformation, Canad. J. Math., 32(4) (1980), 908-923.
  • [11.]H. Usami, Some oscillation theorems for a class of quasilinear elliptic equations, Ann. Math. Pura. Appl., 175 (1998), 277-283.
  • [12.]D. Willett, On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math., 21 (1969), 175-194.
  • [13.]J. S. W. Wong, Oscillation and nonoscillation of solution of second order linear differential equations with integrable coefficients, Trans. Amer. Math. Soc., 114 (1969), 197-215.
  • [14.]Z. Xu and H. Xing, Oscillation criteria of Kamenev-type for PDE with $p-$Laplacian, Appl. Math. Comput., 145 (2003), 735-745.
  • [15.]Z. Xu, B. Jia and S. Xu, Averaging techniques and oscillation of quasilinear elliptic equations, Ann. Polon. Math., 81(1) (2004), 45-54.
  • [16.]Z. Xu and H. Xing, Oscillation criteria for PDE with $p-$Laplacian involving general means, Annali. Math., 184(3) (2005), 395-406.