Local Lipschitz continuity for p,q−PDEs with explicit u−dependence

This article is dedicated to Emmanuele Di Benedetto, great mathematician, colleague, friend. In the spirit to treat a subject that in the last years attracted the interest of several mathematicians, and the attention of Emmanuele too, in this paper we give a first approach to the local Lipschitz continuity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type Σn i=1 ∂ ∂xi ai (x, u,Du) = b (x, u,Du) under p, q−growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable.