The prescribed scalar curvature problem for polyharmonic operator

We consider the following prescribed curvature problem involving polyharmonic operator: $$\begin{aligned} D_mu=Q(|y'|,y'')u^{m^*-1}, \;u>0, \; u \in {\mathcal {H}}^{m}({\mathbb {S}}^{N}), \end{aligned}$$ where $$m^*=\frac{2N}{N-2m},\; N\ge 4m+1$$ , $$m \in {\mathbb {N}}_+$$ , $$(y',y'') \in {\mathbb {R}}^{2} \times {\mathbb {R}}^{N-2}$$ , and $$Q(|y'|,y'')$$ is a bounded nonnegative function in $${\mathbb {R}}^{+} \times {\mathbb {R}}^{N-2}$$ . $${\mathbb {S}}^N$$ is the unit sphere with induced Riemannian metric g, $$D_m$$ is the polyharmonic operator given by $$D_m=\prod _{k=1}^m(-\Delta _g+\frac{1}{4}(N-2k)(N+2k-2)),$$ where $$\Delta _g$$ is the Laplace–Beltrami operator on $${\mathbb {S}}^N$$ . By using a finite reduction argument and local Pohozaev-type identities for polyharmonic operator, we prove that if $$N \ge 4m+1$$ and $$Q(r,y'')$$ has a stable critical point $$(r_0,y_0'')$$ , then the above problem has infinitely many solutions, whose energy can be arbitrarily large.