The Leray-Lions existence theorem under general growth conditions

We prove an existence (and regularity) result of weak solutions $u\in W_0^{1,p}\left(\Omega \right)\cap W_{\mathrm{loc}}^{1,q}\left(\Omega \right)$, to a Dirichlet 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 $p=q$.