Entropy solutions of elliptic equation from two phase problems

The Dirichlet problem of elliptic equation from two phase problems, $b(u)-div({|\mathrm{\nabla }u)|}^{p(x)-2}\mathrm{\nabla }u+\mu (x){\left|\mathrm{\nabla }u\right|}^{q(x)-2}\mathrm{\nabla }u)=f(x)$ $b(u)-\operatorname{div}\left (\left |\nabla u)\right |^{p(x) - 2} \nabla u+\mu (x)\left |\nabla u\right |^{q(x) - 2}\nabla u\right )=f(x)$ in Ω, $u=0$ $u=0$ on ∂Ω is considered, where Ω is a bounded domain with a smooth boundary in ${\mathbb{R}}^N$ $\mathbb{R}^{N}$ , b is a continuous and nondecreasing function. By the theory of the weighted variable Sobolev space, the existence and uniqueness of an entropy solution for $L^1$ $L^{1}$ -data f are proved.