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 Lu = f(x; u,∇Lu) on ω ⊂ℝN; where f is such that f(x; t; ξ) t ≥ 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.