Consider the following Hénon type system: 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |x|^{\alpha } v^{p},\;\;\; & \text {in } \;\; B_1(0),\\ -\Delta v = |x|^{\alpha } u^{q},\;\;\; & \text {in }\;\; B_1(0),\\ u = v= 0, & \text {on } \partial B_1(0), \end{array}\right. } \end{aligned}\) where \(B_1(0)\subset \mathbb {R}^N\) is the unit ball centered at the origin, \(N\ge 3\) , \(\alpha > 0\) and \(p, q>1\) satisfy: \( \dfrac{1}{p+1} + \dfrac{1}{q+1} > \dfrac{N-2}{N}. \) It is shown that the ground state solution of Eq. 0.1 exists for each \(\alpha >0\) and satisfies the foliated Schwartz symmetry property when \(\alpha \) is large. In this paper, we first investigate the asymptotic behavior of the ground state solution \( ( u_\alpha , v_\alpha )\) for Eq. 0.1, as \(\alpha \rightarrow +\infty \) . Then we analyse the profile of the solutions with one peak or multiple peaks.
