We consider the following prescribed curvature problem involving polyharmonic operator:

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.

我们考虑以下涉及多谐算子的规定曲率问题：

其中\（m ^ * = \ frac {2N} {N-2m}，\; N \ ge 4m + 1 \），\（m \ in {\ mathbb {N}} _ + \），\（（y '，y''）\ in {\ mathbb {R}} ^ {2} \ times {\ mathbb {R}} ^ {N-2} \）和\（Q（| y'|，y'' ）\）是\（{\ mathbb {R}} ^ {+} \ times {\ mathbb {R}} ^ {N-2} \）中的有界非负函数。\（{\ mathbb {S}} ^ N \）是具有诱导黎曼度量g的单位球面，\（D_m \）是\（D_m = \ prod _ {k = 1} ^ m（- \ Delta _g + \ frac {1} {4}（N-2k）（N + 2k-2）），\）其中\（\ Delta _g \）是\（{\ mathbb {S} } ^ N \）。通过对多谐波算子使用有限约简参数和局部Pohozaev型恒等式，我们证明了如果\（N \ ge 4m + 1 \）和\（Q（r，y''）\）具有稳定的临界点\（ （r_0，y_0''）\），则上述问题具有无限多个解决方案，其能量可以任意大。