A Liouville comparison principle for solutions ofsingular quasilinear elliptic second-order partial differentialinequalities

Bernd Kawohl; Vasilii Kurta

doi:10.3934/cpaa.2011.10.1747

Article Contents

2011, Volume 10, Issue 6: 1747-1762. Doi: 10.3934/cpaa.2011.10.1747

This issue Previous Article Blow-up rates of large solutions for semilinear elliptic equations Next Article Theory of the NS-$\overline{\omega}$ model: A complement to the NS-$\alpha$ model

A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities

Bernd Kawohl^1, and
Vasilii Kurta^2,

1.
Mathematisches Institut, Universität zu Köln, 50923 Köln
2.
Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, Michigan 48107-8604

Received: March 2011

Revised: April 2011

Published: November 2011

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Abstract

We compare entire weak solutions $u$ and $v$ of quasilinear partial differential inequalities on $R^n$ without any assumptions on their behaviour at infinity and show among other things, that they must coincide if they are ordered, i.e. if they satisfy $u\geq v$ in $R^n$. For the particular case that $v\equiv 0$ we recover some known Liouville type results. Model cases for the equations involve the $p$-Laplacian operator for $p\in[1,2]$ and the mean curvature operator.

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Mathematics Subject Classification: Primary: 35B53 35J62; Secondary: 35B51, 35J75, 35J92, 35J93.

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