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A new variational method for the p (x)-Laplacian equation

Published online by Cambridge University Press:  17 April 2009

Marek Galewski
Affiliation:
Faculty of Mathematics, University of Lodz, Banacha 22, 90–238 Lodz, Poland, e-mail: galewski@math.uni.lodz.pl
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Using a dual variational method we shall show the existence of solutions to the Dirichlet problem without assuming Palais-Smale condition.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Dinca, G. and Jeblean, P., ‘Some existence results for a class of nonlinear equations involving a duality mapping’, Nonlinear Anal. 46 (2001). 347–363.Google Scholar
[2]Ekeland, I. and Temam, R., Convex analysis and variational problems (North-Holland, Amsterdam, 1976).Google Scholar
[3]Fan, X.L. and Zhao, D., ‘Sobolev embedding theorems for spaces W k, p (x)(Ω)’, J. Math. Anal. Appl. 262 (2001), 749760.CrossRefGoogle Scholar
[4]Fan, X.L. and Zhao, D., ‘Existence of solutions for p (x)- Lapacian Dirichlet problem’, Nonlinear Anal. 52 (2003), 18431852.Google Scholar
[5]Fan, X.L. and Zhao, D., ‘On the spaces L p (x)(Ω) and W k, p (x)(Ω)J. Math. Anal. Appl. 263 (2001), 424446.CrossRefGoogle Scholar
[6]El Hamidi, A., ‘Existence results to elliptic systems with nonstandart growth conditions’, J. Math. Anal. Appl. 300 (2004), 3042.CrossRefGoogle Scholar
[7]Morrey, Ch.B., Multiple integrals in the calculus of variations (Springer-Verlag, Berlin, 1966).Google Scholar
[8]Nowakowski, A. and Rogowski, A., ‘On the new variational principles and duality for periodic solutions of Lagrange equations with superlinear nonlinearities’, J. Math. Anal. Appl. 264 (2001), 168181.CrossRefGoogle Scholar
[9]Ruzicka, M., Electrorheological fluids: Modelling and mathematical theory, Lecture Notes in Mathematics 1748 (Springer-Verlag, Berlin, 2000).CrossRefGoogle Scholar
[10]Zhikov, V.V., ‘Averaging of functionals of the calculus of variations and elasticity theory’, Math. USSR-Izv. 29 (1987), 3366.CrossRefGoogle Scholar