The Leray-Lions existence theorem under general growth conditions

We prove an existence (and regularity) result of weak solutions u is an element of W 1,p problem for a second order elliptic equation in divergence form, under general and p, q-growth conditions of the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with q = p. We found a way to treat the 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.