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On Existence and Stability of Solutions to Elliptic Systems with Generalised Growth

Published online by Cambridge University Press:  17 April 2009

Marek Galewski  and
Marek Plócienniczak
Marek Galewski
Affiliation:
Faculty of Mathematics and Computer Science, University of Lódź, Banacha 22, 90–238 Lódź, Poland, e-mail: galewski@math.uni.lodz.pl, plo@math.uni.lodz.pl
Marek Plócienniczak
Affiliation:
Faculty of Mathematics and Computer Science, University of Lódź, Banacha 22, 90–238 Lódź, Poland, e-mail: galewski@math.uni.lodz.pl, plo@math.uni.lodz.pl
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We are concerned with existence and stability of solutions for system of equations with generalised p(x) and m(x)—Laplace operators and where the nonlinearity satisfies some local growth conditions. We provide a variational approach that is based on investigation of the primal and the dual action functionals. As a consequence we consider the dependence of the the system on functional parameters.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 3 , December 2007 , pp. 453 - 470
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Adams, R.A., Sobolev spaces (Academic Press, New York, 1975).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. App. 262 (2001), 749–760.CrossRefGoogle Scholar
[4]Fan, X.L. and Zhang, H., ‘Existence of solutions for p(x)-Laplacian dirichlet problem’, Nonlin. Anal. 52 (2003), 1843–1852.CrossRefGoogle Scholar
[5]Fan, X.L. and Zhao, D., ‘On the spaces L P(x)(Ω) and W k, p(x)(Ω)’, J. Math. Anal. App. 263 (2001), 424–446.CrossRefGoogle Scholar
[6]Galewski, M., ‘New variational method for p(x)- Laplacian equation’, Bull. Austral. Math. Soc. 72 (2005), 53–65.CrossRefGoogle Scholar
[7]Galewski, M., ‘On the continuity of the Nemytskij operator between the spaces L p1(x) and L p2(x)’, Georgian Math. J. 13 (2006), 261–265.CrossRefGoogle Scholar
[8]Galewski, M. and Plócienniczak, M., ‘On the nonlinear Dirichlet problem with p(x)-Laplacian’, Bull. Austral. Math. Soc. 75 (2007), 381–395.CrossRefGoogle Scholar
[9]El Hamidi, A., ‘Existence results to elliptic systems with nonstandart growth conditions’, J. Math. Anal. Appl. 300 (2004), 30–42.CrossRefGoogle Scholar
[10]Idczak, D., ‘Stability in semilinear problems’, J. Differential Equations 162 (2000), 64–90.CrossRefGoogle Scholar
[11]Ruzicka, M., Electrorheological fluids: Modelling and mathematical theory, Lecture Notes in Mathematics 1748 (Springer-Verlag, Berlin, 2000).CrossRefGoogle Scholar
[12]Walczak, S., ‘On the continuous dependence on parameters of solutions of the Dirichlet problem. Part I. Coercive Case, Part II. The Case of Saddle Points’, Acad. Roy. Belg. Bull. Cl. Sci. 6 (1995), 247–273.Google Scholar
[13]Walczak, S., ‘Continuous dependence on parameters and boundary data for nonlinear P.D.E. coercive case’, Differential Integral Equations 11 (1998), 35–46.CrossRefGoogle Scholar
[14]Zhikov, V.V., ‘Averaging of functionals of the calculus of variations and elasticity theory’, Math. USSR Izv. 29 (1987), 33–66.CrossRefGoogle Scholar