Existence and regularity of entropy solutions for strongly nonlinear p(x)-elliptic equations

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