Regularity for Nonuniformly Elliptic Equations with p,q-Growth and Explicit x,u-Dependence

We are interested in the regularity of weak solutions u to the elliptic equation in divergence form; precisely in their local boundedness and their local Lipschitz continuity under general growth conditions, the so called p,q-growth conditions. We found a unique set of assumptions to get all these regularity properties at the same time; in the meantime we also found the way to treat a more general context, with explicit dependence on (x,u), other than on the gradient variable xi=Du; these aspects require particular attention due to the p,q-context, with some differences and new difficulties compared to the standard case p=q.