We study the non-linear fractional elliptic problem \( (-\Delta )^s u = (1 + |x|^\alpha ) |u|^{p-1} u \quad \text {in } \mathcal {H}, \) where \(\mathcal {H} = \mathbb {R}^n\) or \(\mathcal {H} = \mathbb {R}^n_+ = \{x = (x', x_n) \in \mathbb {R}^{n-1} \times \mathbb {R} : x_n > 0\}\) with Dirichlet boundary conditions on \(\mathbb {R}^n_-\) . Here, \(n \ge 2s\) , \(p > 1\) , \(\alpha > -2s\) , and \(0< s < 1\) . This equation incorporates both an autonomous term, \(|u|^{p-1}u\) , and a non-autonomous perturbation, \(|x|^\alpha |u|^{p-1}u\) . Our goal is to analyze the influence of this potential \((1 + |x|^\alpha )\) to classify regular stable solutions, which may be unbounded or sign-changing. We prove the nonexistence of nontrivial stable solutions (respectively, solutions stable outside a compact set) for all \(p > 1\) . Motivated by [14, 21, 23], we also establish a monotonicity formula to study the supercritical case and further characterize the qualitative behavior of solutions.
