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

Lei Qiao

doi:10.3934/dcdsb.2017099

Article Contents

2017, Volume 22, Issue 6: 2321-2338. Doi: 10.3934/dcdsb.2017099

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

Lei Qiao

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

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

Received: May 31, 2016

Revised: October 31, 2016

Published: May 2017

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

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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.

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Mathematics Subject Classification: Primary:35J05, 35J10;Secondary:35J25, 35J67.

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