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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 \[ \left \{ \begin {array}{ll} -\Delta u +\lambda u=\mu _1 |u|^{2p}u+\beta _1 |v|^{q_1} |u|^{p_1-1}u,~~x\in \Omega ,\\ -\Delta v +\lambda v=\mu _2 |v|^{2p}v+\beta _2 |u|^{q_2} |v|^{p_2-1}v,~~x\in \Omega ,\\ u=v=0~~\hbox {on}~ \partial \Omega ,\\ \end {array} \right .\] 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
MR Author ID: 1078772
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
MR Author ID: 635770
Email: sjpeng@mail.ccnu.edu.cn

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