A priori estimates and reduction principles for quasilinear elliptic problems and applications

Abstract

Variants of Kato's inequality are proved for general quasilinear elliptic operators $L$. As an outcome we show that, dealing with Liouville theorems for coercive equations of the type \begin{equation*} Lu = f(x,u,{\nabla_{\!\!L}} u) \quad on\ \Omega\subset{{\mathbb{R}}^N} , \end{equation*} where $f$ is such that $f(x,t,\xi) \,t\ge 0$, the assumption that the possible solutions are nonnegative involves no loss of generality. Related consequences such as comparison principles and a priori bounds on solutions are also presented. An underlying structure throughout this work is the framework of Carnot groups.