On viscosity and weak solutions for non-homogeneous p-Laplace equations

In this manuscript we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower order term depending on $x$, $u$ and $\mathrm{\nabla }u$. More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending previous results by Juutinen, Lindqvist and Manfredi on the homogeneous case, and Julin and Juutinen for a linear right hand side. Moreover, we provide a converse statement in the full case under extra assumptions on the regularity of the solutions.