Local boundedness of weak solutions to elliptic equations with $p,q-$growth

This article is dedicated to Giuseppe Mingione for his ${50}^{th}$ birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the local boundedness of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under $p,q-$growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on $u$, other than on its gradient $Du$ and on the $x$ variable.