Advances in Differential Equations

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

Lorenzo D'Ambrosio and Enzo Mitidieri

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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

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

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B45: A priori estimates 35B51: Comparison principles 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35J62: Quasilinear elliptic equations 35J70: Degenerate elliptic equations 35R03: Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc.


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.

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