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August  2017, 22(6): 2321-2338. doi: 10.3934/dcdsb.2017099

## Asymptotic behaviors of Green-Sch potentials at infinity and its applications

 School of of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China

Received  June 2016 Revised  November 2016 Published  March 2017

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant Nos. 11301140, U1304102)

The first aim in this paper is to deal with asymptotic behaviors of Green-Sch potentials in a cylinder. As an application we prove the integral representation of nonnegative weak solutions of the stationary Schrödinger equation in a cylinder. Next we give asymptotic behaviors of them outside an exceptional set. Finally we obtain a quantitative property of rarefied sets with respect to the stationary Schrödinger operator at $+\infty$ in a cylinder. Meanwhile we show that the reverse of this property is not true.

Citation: Lei Qiao. Asymptotic behaviors of Green-Sch potentials at infinity and its applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2321-2338. doi: 10.3934/dcdsb.2017099
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