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In this article, we are concerned with the following prescribed curvature problem involving polyharmonic operator onSN{{\mathbb{S}}}^{N}:Dmu=K(∣y∣)um∗−1,u&gt;0inSN,u∈Hm(SN),{D}^{m}u=K\left(| y| ){u}^{{m}^{\ast }-1},\hspace{1.0em}u\gt 0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{S}}}^{N},\hspace{1.0em}u\in {H}^{m}\left({{\mathbb{S}}}^{N}),whereK(∣y∣)K\left(| y| )is a positive function,m∗=2NN−2m{m}^{\ast }=\frac{2N}{N-2m}is the Sobolev embedding critical exponent,N&gt;2m+2N\gt 2m+2.Dm{D}^{m}is the2m2morder differential operator given byDm=∏l=1m−Δg+14(N−2l)(N+2l−2),{D}^{m}=\mathop{\prod }\limits_{l=1}^{m}\left(-{\Delta }_{g}+\frac{1}{4}\left(N-2l)\left(N+2l-2)\right),whereΔg{\Delta }_{g}is the Laplace-Beltrami operator onSN{{\mathbb{S}}}^{N},SN{{\mathbb{S}}}^{N}is the unit sphere with Riemann metricgg. We first establish two kinds of local Pohozaev identities for polyharmonic operator, then we prove that the positive bubbling solution constructed in the study of Guo and Li is non-degenerate.

Non-degeneracy of bubble solutions for higher order prescribed curvature problem