  We prove an existence 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 $\left(
x,u\right) $, 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 $p=q$.
