ANZIAM J. 49 (2007), no. 1, pp. 75–83.
A note on the stability and the approximation of solutions for a Dirichlet problem with p(x)-Laplacian
Marek Galewski
Faculty of Mathematics and Computer Science
University of Lodz
Banacha 22
90-238 Lodz
Poland
galewski@math.uni.lodz.pl.
Received 1 September 2006; revised 11 June 2007

Abstract

We show the stability results and Galerkin-type approximations of solutions for a family of Dirichlet problems with nonlinearity satisfying some local growth conditions.

Download the article in PDF format (size 120 Kb)

2000 Mathematics Subject Classification: primary 35A15; secondary 35B35, 65N30
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2378150 Z'blatt-MATH: pre05243892

References

  1. R. A. Adams, Sobolev Spaces (Academic Press, New York, 1975). MR450957
  2. S. N. Antontsev and S. I. Shmarev, “Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of the solutions”, Nonlinear Anal. 65 (2006) 728–761. MR2232679
  3. J. Chabrowski and Y. Fu, “Existence of solutions for p(x)-Laplacian problem on a bounded domain”, J. Math. Anal. Appl. 306 (2005) 604–618. MR2136336
  4. I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976). MR463994
  5. X. L. Fan and H. Zhang, “Existence of solutions for p(x)-Laplacian Dirichlet problem”, Nonlinear Anal. 52 (2003) 1843–1852. MR1954585
  6. X. L. Fan and D. Zhao, “On the spaces L^{p}(x)(Ω) and W^{k},p(x)(Ω)”, J. Math. Anal. Appl. 263 (2001) 424–446. MR1866056
  7. X. L. Fan and D. Zhao, “Sobolev embedding theorems for spaces W^{k},p(x)(Ω)”, J. Math. Anal. Appl. 262 (2001) 749–760. MR1859337
  8. M. Galewski, “On the existence and stability of solutions for Dirichlet problem with p(x)-Laplacian”, J. Math. Anal. Appl. 326 (2007) 352–362. MR2277787
  9. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasi-linear elliptic equations (Academic Press, New York, 1968). MR244627
  10. M. Růžička, Electrorheological fluids: Modelling and Mathematical Theory, Volume 1748 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 2000). MR1810360
  11. V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory”, Math. USSR Izv. 29 (1987) 33–66. MR864171