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 Schrodinger equation and the Martin boundary of the product of Riemannian manifolds. For their sharpness, we show that the Martin compactification of R-2 for some Schrodinger equation is so much distorted near infinity that no product structures remain.
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