We consider second order elliptic equations in divergence form \(\begin{aligned} \sum _{i=1}^{n}\frac{\partial }{\partial x_{i}}a^{i}\left( x,u,Du\right) =b\left( x,u,Du\right) ,\;\;\;\;\;x\in \Omega , \end{aligned}\) where \(\Omega \) is a bounded open set in \(\mathbb {R}^{n}\) and \(u:\Omega \rightarrow \mathbb {R}\) . Our aim is to give conditions on the vector field \( a\left( x,u,Du\right) =\left( a^{i}\left( x,u,Du\right) \right) _{i=1,\ldots ,n}\) and on the right hand side \(b\left( x,u,Du\right) \) in order to obtain the global boundedness in \(\overline{\Omega }\) of weak solutions u to the Dirichlet problem associated to the previous differential equation, when a boundary condition \(u=u_{0}\in L^{\infty }\left( \Omega \right) \) has been fixed on \(\partial \Omega \) . We do not assume structure conditions on the vector field \(a\left( x,u,Du\right) \) , nor sign assumptions on \(b\left( x,u,Du\right) \) ; 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 \(\frac{q}{p}\) , but nothing more than \(1\le p\le q\,\) .
