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

Abstract

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.