Equivalence of solutions for non-homogeneous p(x)-Laplace equations?

We establish the equivalence between weak and viscosity solutions for non-homogeneous p(x)-Laplace equations with a right-hand side term depending on the spatial variable, the unknown, and its gradient. We employ inf- and sup-convolution techniques to state that viscosity solutions are also weak solutions, and comparison principles to prove the converse. The new aspects of the p(x)Laplacian compared to the constant case are the presence of log-terms and the lack of the invariance under translations.