In this work, we explore the asymptotic behaviors of positive solutions to the \(\sigma _k\) -Yamabe equation. Extending Han and Li’s previous work on the Yamabe equation, we demonstrate that for every approximate solution \({\widetilde{w}}\) of a specified order, there exists a corresponding solution w that closely approximates \({\widetilde{w}}\) . Our study further presents a concrete method for constructing these approximate solutions. By appropriately perturbing the radial solution, we can consistently obtain an approximate solution with a designated order. This approach offers significant insights into the characteristics and behaviors of solutions near isolated singular points.
