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A note on the stability and the approximation of solutions for a Dirichlet problem with p(x)-Laplacian

Published online by Cambridge University Press:  17 February 2009

Marek Galewski
Marek Galewski
Affiliation:
Faculty of Mathematics and Computer Science University of LodzBanacha 22 90–238 Lodz Poland email:galewski@math.uni.lodz.pl.
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We show the stability results and Galerkin-type approximations of solutions for a family of Dirichlet problems with nonlinearity satisfying some local growth conditions. 2000 Mathematics subject classification: primary 35A15; secondary 35B35, 65N30. Keywords and phrases: p(x)-Laplacian, duality, variational method, stability of solutions, Galerkin-type approximations.

Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 1 , July 2007 , pp. 75 - 83
Copyright
Copyright © Australian Mathematical Society 2007

References

reference

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