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

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