Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T02:31:16.307Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected]
Marek Plócienniczak
Affiliation:
Faculty of Mathematics and Computer Science, University of Lódź, Banacha 22, 90–238 Lódź, Poland, e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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