Synchronized vector solutions to an elliptic system

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.