Summary: In this paper we study solutions, possibly unbounded and sign-changing, of the following equation \[ -\Delta_{\lambda} u=|x|_{\lambda}^a |u|^{p-1}u \quad\text{in } \mathbb {R}^n, \] where \(n\geq 1\), \(p>1\), \(a \geq 0\) and \(\Delta_{\lambda}\) is a strongly degenerate elliptic operator, the functions \(\lambda =(\lambda_1,\dots, \lambda_k) : \; \mathbb {R}^n \rightarrow \mathbb {R}^k\) satisfies some certain conditions, and \(|.|_{\lambda}\) the homogeneous norm associated to the \(\Delta_{\lambda}\)-Laplacian. We prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of \(\mathbb {R}^n\). First, we establish the standard integral estimates via stability property to derive the nonexistence results for stable solutions. Next, by mean of the Pohozaev identity, we deduce the Liouville-type theorem for solutions stable outside a compact set.