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

Bernd Kawohl,Vasilii Kurta

doi:10.3934/cpaa.2011.10.1747

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.

Highlights

Introduction and DefinitionsThis work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear elliptic second-order differential inequalities of the formsA(u) + |u|q−1u ≤ A(v) + |v|q−1v (1) and−A(u) + |u|q−1u ≤ −A(v) + |v|q−1v (2)on Rn, where n ∈ N, 0 < q ∈ R, and the operator A(w) belongs to the class of so-called α-monotone operators
We present algebraic inequalities which imply immediately that the pLaplacian operator ∆p(w) and its modification ∆p(w) satisfy the α-monotonicity condition for α = p and 1 < p ≤ 2
In Theorems 1–5 we formulate our results for solutions of inequality (1) and in Theorem 6 for solutions of (2)

Summary

Introduction

Introduction and DefinitionsThis work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear elliptic second-order differential inequalities of the formsA(u) + |u|q−1u ≤ A(v) + |v|q−1v (1) and−A(u) + |u|q−1u ≤ −A(v) + |v|q−1v (2)on Rn, where n ∈ N, 0 < q ∈ R, and the operator A(w) belongs to the class of so-called α-monotone operators. 1. Suppose that the operator A(w) is α-monotone and (u, v) is an entire weak solution of inequality (1) on Rn such that u(x) ≥ v(x).

Results

Conclusion