Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 68, pp. 1-27.

Title: Existence and regularity of entropy solutions for strongly
       nonlinear p(x)-elliptic equations
        
Authors: Elhoussine Azroul (Univ. of Fez, Morocco)
         Hassane Hjiaj     (Univ. of Fez, Morocco)
         Abdelfattah Touzani (Univ. of Fez, Morocco)

Abstract:
 This article is devoted to study the existence of solutions for
 the  strongly nonlinear p(x)-elliptic problem
 $$\displaylines{
 - \operatorname{div}  a(x,u,\nabla u) + g(x,u,\nabla u)
 = f- \operatorname{div}  \phi(u) \quad \text{in } \Omega, \cr
  u  =   0 \quad \text{on }  \partial\Omega,
 }$$
 with $ f\in L^1(\Omega) $ and $ \phi \in  C^{0}(\mathbb{R}^{N})$,
 also we will give some  regularity results for these solutions.

Submitted May 22, 2012. Published March 08, 2013.
Math Subject Classifications: 35J20, 35J25, 35J60.
Key Words: Sobolev spaces with variable exponents; entropy solutions;
           strongly nonlinear elliptic equations; boundary value problems.