Maximum principles for generalized solutions of quasi-linear elliptic equations (Q1887373)

This article considers quasi-linear equation \[ -\text{ div}A(x,u,\nabla u) +B(x,u,\nabla u)=0 \quad \text{ in }\;G, \] where \(G\) is a bounded domain in \(\mathbb R^{n}\), \(A(x,u,\xi)\) and \(B(x,u,\xi)\) are ``Carathéodory'' functions, satisfying certain elliptic and growth conditions. The authors prove that if a bounded generalized solution \(u\) is not a constant, then \[ \text{ ess\;sup}_{G'}u< \text{ ess\;sup}_{G}u, \quad\forall\;G'\subset\subset G. \]