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

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

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We establish the equivalence between weak and viscosity solutions for non-homogeneous $ p(x) $-Laplace equations with a right-hand side term depending B-100 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 Online 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.