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Synchronized vector solutions to an elliptic system


Authors: Qihan He and Shuangjie Peng
Journal: Proc. Amer. Math. Soc. 144 (2016), 4055-4063
MSC (2010): Primary 58J10; Secondary 58J20
DOI: https://doi.org/10.1090/proc/13160
Published electronically: April 27, 2016
MathSciNet review: 3513560
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Abstract: In this paper, we establish a relationship between the elliptic system

$\displaystyle \left \{ \begin {array}{ll} -\Delta u +\lambda u=\mu _1 \vert u\v... ...ga ,\vspace {2mm}\\ u=v=0~~\hbox {on}~ \partial \Omega ,\\ \end{array} \right .$

and its corresponding single elliptic problem, where $ \lambda \in \mathbb{R}$, $ \beta _i>0, \mu _i<0, p_i,q_i\ge 0, 1<p_i+q_i =2p+1$ for $ i=1,2$, and $ \Omega \subset \mathbb{R}^N\,(N\ge 1)$ can be a bounded or unbounded domain. By using this fact, we can obtain many results on the existence, non-existence and uniqueness of classical vector solutions to this system via the related single elliptic problem.

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Additional Information

Qihan He
Affiliation: School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, People’s Republic of China
Email: heqihan277@163.com

Shuangjie Peng
Affiliation: School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, People’s Republic of China
Email: sjpeng@mail.ccnu.edu.cn

DOI: https://doi.org/10.1090/proc/13160
Received by editor(s): October 10, 2014
Received by editor(s) in revised form: June 22, 2015
Published electronically: April 27, 2016
Communicated by: Nimish A. Shah
Article copyright: © Copyright 2016 American Mathematical Society

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