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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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