## Advances in Differential Equations

### 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.

#### Article information

Source
Adv. Differential Equations, Volume 17, Number 9/10 (2012), 935-1000.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355702928

Mathematical Reviews number (MathSciNet)
MR2985680

Zentralblatt MATH identifier
1273.35138

#### Citation

D'Ambrosio, Lorenzo; Mitidieri, Enzo. A priori estimates and reduction principles for quasilinear elliptic problems and applications. Adv. Differential Equations 17 (2012), no. 9/10, 935--1000. https://projecteuclid.org/euclid.ade/1355702928