The prescribed scalar curvature problem for polyharmonic operator

We consider the following prescribed curvature problem involving polyharmonic operator: D_mu=Q(|y'|,y”)u^m^*-1, u>0, u ∈ℋ^m(𝕊^N), where m^*=2N/N-2m, N≥ 4m+1 , m ∈ℕ_+ , (y',y”) ∈ℝ^2×ℝ^N-2 , and Q(|y'|,y”) is a bounded nonnegative function in ℝ^+×ℝ^N-2 . 𝕊^N is the unit sphere with induced Riemannian metric g , D_m is the polyharmonic operator given by D_m=∏ _k=1^m(-Δ _g+1/4(N-2k)(N+2k-2)), where Δ _g is the Laplace–Beltrami operator on 𝕊^N . By using a finite reduction argument and local Pohozaev-type identities for polyharmonic operator, we prove that if N ≥ 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.