Global boundedness of weak solutions to a class of nonuniformly elliptic equations

We consider second order elliptic equations in divergence form ∂∂xi ai (x, u, Du) = b (x, u, Du) , x ∈ , where is a bounded open set in Rn and u : → R. Our aim is to give conditions on the vector field a (x, u, Du) = ai (x, u, Du) i=1,...,n and on the right hand side b (x, u, Du) in order to obtain the global boundedness in of weak solutions u to the Dirichlet problem associated to the previous differential equation, when a boundary condition u = u0 ∈ L∞ ( ) has been fixed on ∂ . We do not assume structure conditions on the vector field a (x, u, Du), nor sign assumptions on b (x, u, Du); we only consider ellipticity and growth conditions on a and b. A main novelty with respect to the literature about this subject is that we assume general p, q−growth conditions for the principal part of the differential equation; however we do not need an upper bound for the ratio q/p , but nothing more than 1 ≤ p ≤ q .