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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Applications of phase plane analysis of a Liénard system to positive solutions of Schrödinger equations


Authors: Jitsuro Sugie and Naoto Yamaoka
Journal: Proc. Amer. Math. Soc. 131 (2003), 501-509
MSC (2000): Primary 35B05, 35J60; Secondary 34C10, 70K05
Published electronically: June 12, 2002
MathSciNet review: 1933341
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Abstract: This paper deals with semilinear elliptic equations in an exterior domain of $\mathbb{R}^N$ with $N \ge 3$. Sufficient conditions are obtained for the equation to have a positive solution which decays at infinity. The main result is proved by means of a supersolution-subsolution method presented by Noussair and Swanson. By using phase plane analysis of a system of Liénard type, a suitable positive supersolution is found out. Asymptotic decay estimation on a solution of the Liénard system gains a positive subsolution. Examples are given to illustrate the main result.


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Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
Email: jsugie@math.shimane-u.ac.jp

Naoto Yamaoka
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
Email: yamaoka@math.shimane-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06681-9
PII: S 0002-9939(02)06681-9
Keywords: Positive solution, Schr\"{o}dinger equation, exterior domain, Li\'enard system
Received by editor(s): September 19, 2001
Published electronically: June 12, 2002
Additional Notes: The first author was supported in part by Grant-in-Aid for Scientific Research 11304008
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2002 American Mathematical Society