A fixed point method for the $ p(\cdot )$-Laplacian

George Dinca

doi:10.1016/j.crma.2009.04.022

Partial Differential Equations

A fixed point method for the

p (\cdot)

-Laplacian
[Une méthode de point fixe pour le

p (\cdot)

-laplacien]

Présenté par : Jean-Pierre Kahane

George Dinca ¹

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

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 757-762.

Résumé
Abstract

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 $W_{0}^{1, p (\cdot)} (Ω)$ 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 ( $p (x) \equiv p = cte. > 1$ ).

A topological method, based on the fundamental properties of the Leray–Schauder degree, is used in proving the existence of a week solution in $W_{0}^{1, p (\cdot)} (Ω)$ to Dirichlet problem

- div ({| \nabla u |}^{p (x) - 2} \nabla u) = f (x, u), x \in Ω, (P)

u = 0, x \in \partial 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) \equiv p = const. > 1

).

Reçu le : 2009-03-26
Accepté le : 2009-03-31
Publié le : 2009-05-14

DOI : 10.1016/j.crma.2009.04.022

Affiliations des auteurs :

George Dinca ¹

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

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     title = {A fixed point method for the $ p(\cdot )${-Laplacian}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {757--762},
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     year = {2009},
     doi = {10.1016/j.crma.2009.04.022},
     language = {en},
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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/

Bibliographie
Cité par

[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 $L^{p (x)} (Ω)$ and $W^{m, p (x)} (Ω)$ , 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 $W_{M}^{k} (Ω)$ , Comment. Math., Volume XXI (1979), pp. 315-324

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