Double-tower Solutions for Higher Order Prescribed Curvature Problem

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 $\tilde{K}(y)>0$ is a radial function, $m^{\ast }=\frac{2N}{N-2m}$ and $D^m$ is $2m$ order differential operator given by \begin{equation*} D^m=\prod_{i=1}^m\left(-Δ_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.