  We consider the following higher order prescribed curvature problem on $
{\mathbb{S}}^N : $ \begin{equation*} D^m \tilde u=\widetilde{K}(y) \tilde
u^{m^{*}-1} \quad \mbox{on} \ {\mathbb {S}}^N, \qquad \tilde u >0 \quad
\mbox{in} \ {\mathbb {S}}^N. \end{equation*} where $\widetilde{K}(y)>0$ is a
radial function, $m^{*}=\frac{2N}{N-2m}$ and $D^m$ is $2m$ order differential
operator given by \begin{equation*}
D^m=\prod_{i=1}^m\left(-\Delta_g+\frac{1}{4}(N-2i)(N+2i-2)\right),
\end{equation*} where $g=g_{{\mathbb{S}}^N}$is the Riemannian metric. We prove
the existence of infinitely many double-tower type solutions, which are
invariant under some non-trivial sub-groups of $O(3),$ and their energy can be
made arbitrarily large.
