Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 144, pp. 1-11.

Title: Weak solutions for anisotropic nonlinear elliptic equations with
       variable exponents

Authors: Blaise Kone (Univ. de Ouagadougo, Burkina Faso)
         Stanislas Ouaro (Univ. de Ouagadougo, Burkina Faso)
	 Sado Traore (Univ. de Ouagadougo, Burkina Faso)
       
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.