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

TL;DR

This paper establishes a Liouville comparison principle for solutions of semilinear elliptic inequalities in the whole space, linking solutions' behavior to a capacity related to the elliptic operator.

Contribution

It introduces a new Liouville comparison principle based on capacity for solutions of elliptic inequalities involving measurable coefficients.

Findings

Liouville comparison principle proved for solutions in Sobolev spaces

Capacity associated with the elliptic operator is key to the principle

Applicable to operators with measurable, locally bounded coefficients

Abstract

We consider semilinear elliptic second-order partial differential inequalities of the form Lu +|u|q-1u < and = Lv +|v|q-1v (*) in the whole space Rn, where n > and = 2, q > 0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are defined, measurable and locally bounded in Rn, and that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of a capacity associated with the operator L for solutions of (*), which are defined and measurable in Rn and which belong locally to a Sobolev-type function space also associated with the operator L.