Comptes Rendus
Partial Differential Equations
A fixed point method for the p()-Laplacian
[Une méthode de point fixe pour le p()-laplacien]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 757-762.

On utilise une méthode topologique, basée sur les propriétés fondamentales du degré de Leray–Schauder, afin de démontrer l'existence d'une solution faible dans W01,p()(Ω) pour le problème de Dirichlet (P). Cette méthode représente une adaptation de celle utilisée par Dinca et al. [G. Dinca, P. Jebelean, Une méthode de point fixe pour le p-laplacien, C. R. Acad. Sci. Paris, Ser. I 324 (1997) 165–168. [1], G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Portugal. Math. 53 (3) (2001) 339–377. [2]] pour le p-laplacien classique (p(x)p=cte.>1).

A topological method, based on the fundamental properties of the Leray–Schauder degree, is used in proving the existence of a week solution in W01,p()(Ω) to Dirichlet problem

div(|u|p(x)2u)=f(x,u),xΩ,(P)
u=0,xΩ.
This method is an adaptation of that used by Dinca et al. [G. Dinca, P. Jebelean, Une méthode de point fixe pour le p-laplacien, C. R. Acad. Sci. Paris, Ser. I 324 (1997) 165–168. [1], G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Portugal. Math. 53 (3) (2001) 339–377. [2]] for Dirichlet problems with classical p-Laplacian (p(x)p=const.>1).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.04.022

George Dinca 1

1 Faculty of Mathematics and Computer Science, 14, Academiei St, 010014 Bucharest, Romania
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George Dinca. A fixed point method for the $ p(\cdot )$-Laplacian. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 757-762. doi : 10.1016/j.crma.2009.04.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.022/

[1] G. Dinca; P. Jebelean Une méthode de point fixe pour le p-laplacien, C. R. Acad. Sci. Paris, Ser. I, Volume 324 (1997), pp. 165-168

[2] G. Dinca; P. Jebelean; J. Mawhin Variational and topological methods for Dirichlet problems with p-Laplacian, Portugal. Math., Volume 53 (2001) no. 3, pp. 339-377

[3] X.L. Fan Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., Volume 339 (2008), pp. 1395-1412

[4] X.L. Fan; Q.H. Zhang Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal., Volume 52 (2003), pp. 1843-1852

[5] X.L. Fan; D. Zhao On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., Volume 263 (2001), pp. 424-446

[6] H. Hudzik The problems of separability, duality, reflexivity and comparison for generalized Orlicz–Sobolev spaces WMk(Ω), Comment. Math., Volume XXI (1979), pp. 315-324

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