The Leray-Lions existence theorem under (p,q)-growth conditions

In this talk I will describe recent results obtained in collaboration with P. Marcellini and E. Mascolo. In particular, we proved an existence result of weak solutions to a Dirichlet problem associated to second order elliptic equations in divergence form satisfying (p,q)-growth conditions. This is a first attempt to extend to (p,q)-growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions.
Our existence result is obtained "via regularity", i.e., by using new local regularity results (boundedness, Lipschitz continuity and higher differentiability) for the weak solutions of the associated equation.