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 as in (1.1), and more precisely in their local boundedness and their local Lipschitz continuity under general growth conditions, the so called \(p,\!q\) $p,q$ -growth conditions, as in (1.2) and (1.3) below. We found a unique set of assumptions to get all of 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 \(\left( x,u\right) \) $\left(x,,,u\right)$ , in addition to the gradient variable \(\xi =Du\) $\xi =Du$ . These aspects require particular attention, due to the \(p,\!q\) $p,q$ -context, with some differences and new difficulties compared to the standard case \(p=q\) $p=q$ .