Regularity for nonuniformly elliptic equations with $p,q-$growth and explicit $x,u-$dependence

Abstract

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 $p=q$.