The main goal here is to classify the stable solutions, or more generally, the solutions that remain stable outside a compact set of \(\mathbb {R}^N\) , of the following k-coupled Hardy-Hénon elliptic system \(\begin{aligned} -\Delta u_i = \sum _{j=1}^k \beta _{ij} |x|^{\alpha } |u_j|^{q+1} |u_i|^{q-1} u_i \quad \text {in } \mathbb {R}^N, \quad i = 1, \dots , k, \end{aligned}\) where \( q \ge 1 \) , \( N > 2 \) , \( \alpha > -2 \) , k is a fixed positive integer, and \( B = [\beta _{ij}]_{i,j=1}^k \) is a real symmetric copositive matrix. We consider the critical Joseph-Lundgren exponent \(p_c\) , introduced in (3), for the nonautonomous case, which is larger than the Sobolev critical exponent \(p_S\) for Hardy-Hénon type equations, given in (2). Thus, by setting \(p = q+1\) , we prove certain Liouville-type results for stable solutions and solutions with finite Morse index when \(3 \le p < p_c\) . To achieve this, we employ techniques such as integral estimates based on the stability condition, the application of a monotonicity formula and a Pohozaev identity, and analysis via a blow-down sequence. Moreover, when \(p \ge p_c\) , we show that the system admits nontrivial stable solutions. Then, we establish sharp results regarding the range of p, addressing both the autonomous and nonautonomous vectorial cases.
