In this paper, we consider the following elliptic system of Hénon type on a bounded domain: \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |v|^{p-1}v, & \hbox { in }B_{1}(0), \\ -\Delta v = |y|^{\alpha }|u|^{q-1}u,& \hbox { in }B_{1}(0), \\ u=v=0, & \hbox { on }\partial B_{1}(0), \end{array}\right. } \end{aligned}\) where \(\alpha >0\) , \(B_1(0)\) is the unit ball in \(\mathbb {R}^{N}\) , \(N\ge 5\) , \(1< p<\frac{N-1}{N-2}<q\) and (p, q) is a pair of positive numbers lying on the critical hyperbola \(\begin{aligned} \begin{aligned} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{aligned} \end{aligned}\) We prove the existence of infinitely many non-radial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. And the most ingredients of the paper are using the Green representation and estimating the Green’s function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.
