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

We consider second order elliptic equations in divergence form \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}a^{i}( x,u,Du) =b( x,u,Du); x\in \Omega, 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 $a(x,u,Du\right) =\left( a^{i}( x,u,Du)) _{i=1,\ldots,n}$ and on the right hand side b(x,u,Du) 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 \textit{structure conditions} on the vector field $a\left( x,u,Du\right) $, 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 \frac{q}{p}, but noting more than 1