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

OCHOA, PABLO; MARÍA MEDINA

Mathematics in Engineering

AIMS PRESS

Año: 2023 vol. 5

2640-3501

We establish the equivalence between weak and viscosity solutions for non-homogeneousp(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 arealso 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 invarianceunder translations.