A Liouville comparison principle for solutions of quasilinear differential inequalities

TL;DR

This paper establishes a Liouville comparison principle for entire weak solutions of certain quasilinear differential inequalities involving $ ext{α}$-monotone operators, extending previous results and applicable to operators like the $p$-Laplacian.

Contribution

It introduces a new Liouville comparison principle for solutions of quasilinear inequalities involving $ ext{α}$-monotone operators, broadening the scope of previous results.

Findings

Proves a Liouville comparison principle for $ ext{α}$-monotone operators.

Extends results to include $p$-Laplacian and similar operators.

Provides conditions under which solutions are trivial or comparable.

Abstract

This work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear differential inequalities of the form $A(u) + |u|^{q-1}u \leq A(v) + |v|^{q-1}v$ on ${\Bbb R}^n$, where $n\geq 1$, $q$ is positive, and the operator $A(w)$ belongs to a class of the so-called $\alpha$-monotone operators. Typical examples of such operators are the $p$-Laplacian and its well-known modifications. The results improve and supplement those in [1].