Comptes Rendus
no. 13-14
Partial Differential Equations
A fixed point method for the p(⋅)-Laplacian
Jean-Pierre Kahane
George Dinca1
1Faculty of Mathematics and Computer Science, 14, Academiei St, 010014 Bucharest, Romania
Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 757-762

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)−2∇u)=f(x,u),x∈Ω,(P)
u=0,x∈∂Ω.
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).

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).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.04.022

George Dinca  1

1 Faculty of Mathematics and Computer Science, 14, Academiei St, 010014 Bucharest, Romania 
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
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[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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