Abstract:

In this paper, we study the following elliptic system $\begin{eqnarray*} \left\{\begin{array}{ll} -\Delta u+\mu_1 u=\frac{p}{p+q}|x|^\alpha u^{p-1}v^q, \ &x\in\Omega, \[3mm] -\Delta v+\mu_2 v=\frac{q}{p+q}|x|^\alpha u^pv^{q-1}, \ &x\in\Omega, \[2mm] u, v>0, ~x\in\Omega, ~~u=v=0, ~~&x\in\partial\Omega, \end{array}\right. \end{eqnarray*} $ where $\Omega\subset\mathbb{R}^N$ is an annulus $N\geq 4$, $\mu_1, \mu_2>0$, $p, q>1$ and $p+q<\frac{2N-2}{N-3}$. By variational method and rescaling method, we prove that the system has many non-radial solutions.

Key words: Elliptic system, Variational method, Non-radial solutions