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

Pages: 1747 - 1762, Volume 10, Issue 6, November 2011

doi:10.3934/cpaa.2011.10.1747       Abstract        References        Full Text (417.4K)       Related Articles

Bernd Kawohl - Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany (email)
Vasilii Kurta - Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, Michigan 48107-8604, United States (email)

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.

Keywords:  Liouville theorem, comparison principle, quasilinear elliptic equation, singular elliptic equation.
Mathematics Subject Classification:  Primary: 35B53 35J62; Secondary: 35B51, 35J75, 35J92, 35J93.

Received: March 2011;      Revised: April 2011;      Published: May 2011.

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