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Existence of semiclassical ground-state solutions for semilinear elliptic systems

Published online by Cambridge University Press:  10 August 2012

Jun Wang ,
Junxiang Xu  and
Fubao Zhang
Jun Wang
Affiliation:
Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu 212013, People's Republic of China (wangjdn2006@21cn.com)
Junxiang Xu
Affiliation:
Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, People's Republic of China (xujun@seu.edu.cn; zhangfubao@seu.edu.cn)
Fubao Zhang
Affiliation:
Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, People's Republic of China (xujun@seu.edu.cn; zhangfubao@seu.edu.cn)
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Abstract

This paper is concerned with the following semilinear elliptic equations of the form

where ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε > 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 142 , Issue 4 , August 2012 , pp. 867 - 895
Copyright
Copyright © Royal Society of Edinburgh 2012

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