Double-tower solutions for higher-order prescribed curvature problem

We consider the following higher-order prescribed curvature problem on \( {\mathbb {S}}^N: \)

$$\begin{aligned} D^m {\tilde{u}}=\widetilde{K}(y) {\tilde{u}}^{m^{*}-1} \quad \text{ on } \ {\mathbb {S}}^N, \qquad {\tilde{u}} >0 \quad {\quad \hbox {in } }{\mathbb {S}}^N. \end{aligned}$$

where \(\widetilde{K}(y)>0\) is a radial function, \(m^{*}=\frac{2N}{N-2m}\), and \(D^m\) is the 2m-order differential operator given by

$$\begin{aligned} D^m=\prod _{i=1}^m\left( -\Delta _g+\frac{1}{4}(N-2i)(N+2i-2)\right) , \end{aligned}$$

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.