## Advances in Differential Equations

### Positive solutions of Schrödinger equations and Martin boundaries for skew product elliptic operators

#### Abstract

We consider positive solutions of elliptic partial differential equations on non-compact domains of Riemannian manifolds. We establish general theorems which determine Martin compactifications and Martin kernels for a wide class of elliptic equations in skew product form, by thoroughly exploiting parabolic Martin kernels for associated parabolic equations developed in [35] and [25]. As their applications, we explicitly determine the structure of all positive solutions to a Schrödinger equation and the Martin boundary of the product of Riemannian manifolds. For their sharpness, we show that the Martin compactification of ${\mathbb R}^2$ for some Schrödinger equation is so much distorted near infinity that no product structures remain.

#### Article information

Source
Adv. Differential Equations, Volume 22, Number 9/10 (2017), 621-692.

Dates
First available in Project Euclid: 27 May 2017

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