Electron. J. Diff. Equ., Vol. 2009(2009), No. 144, pp. 1-11.

Weak solutions for anisotropic nonlinear elliptic equations with variable exponents

Blaise Kone, Stanislas Ouaro, Sado Traore

Abstract:
We study the anisotropic boundary-value problem
$$\displaylines{ -\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}}a_{i}(x,\frac{\partial}{\partial x_{i}}u)=f \quad \hbox{in } \Omega, \cr u=0 \quad\hbox{on }\partial \Omega, }$$
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ $(N\geq 3)$. We obtain the existence and uniqueness of a weak energy solution for $f\in L^{\infty}(\Omega)$, and the existence of weak energy solution for general data $f$ dependent on $u$.

Submitted February 10, 2008. Published November 12, 2009.
Math Subject Classifications: 35J20, 35J25, 35D30, 35B38, 35J60.
Key Words: Anisotropic Sobolev spaces; weak energy solution; variable exponents; electrorheological fluids.

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Blaise Kone
Laboratoire d'Analyse Mathématique des Equations (LAME)
Institut Burkinabé des Arts et Métiers, Université de Ouagadougou
03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso
email: leizon71@yahoo.fr
Stanislas Ouaro
Laboratoire d'Analyse Mathématique des Equations (LAME)
Institut Burkinabé des Arts et Métiers, Université de Ouagadougou
03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso
email: souaro@univ-ouaga.bf, ouaro@yahoo.fr
Sado Traore
Laboratoire d'Analyse Mathématique des Equations (LAME)
Institut Burkinabé des Arts et Métiers, Université de Ouagadougou
01 BP 1091 Bobo-Dioulasso 01,
email: sado@univ-ouaga.bf

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