Monotonicity of non-Liouville property for positive solutions of skew product elliptic equations
Minoru Murata, Tetsuo Tsuchida · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 2019
Abstract We consider a second-order elliptic operatorLin skew product of an ordinary differential operatorL1on an interval (a,b) and an elliptic operator on a domainD2of a Riemannian manifold such that the associated heat kernel is intrinsically ultracontractive. We give criteria for criticality and subcriticality ofLin terms of a positive solution having minimal growth atη(η = a,b) to an associated ordinary differential equation. In the subcritical case, we explicitly determine the Martin compactification and Martin kernel forLon the basis of [24]; in particular, the Martin boundary overηis either one point or a compactification ofD2, which depends on whether an associated integral nearηdiverges or converges. From this structure theorem we show a monotonicity property that the Martin boundary overηdoes not become smaller as the potential term ofL1becomes larger nearη.