In this paper, the prescribed
\(\sigma \)-curvature problem
$$\begin{aligned} P_{\sigma }^{g_0} u={\tilde{K}}(x)u^{\frac{N+2\sigma }{N-2\sigma }}, x\in {\mathbb {S}}^N,u>0 \end{aligned}$$
is considered. When
\({\tilde{K}}(x)\) is some axis symmetric function on
\({\mathbb {S}}^N\), by using singular perturbation method, it is proved that this problem possesses infinitely many non-radial solutions for
\(0<\sigma \le 1\) and
\(N> 2\sigma +2\).