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

Abstract

Abstract In this article, we are concerned with the following prescribed curvature problem involving polyharmonic operator on S N {{\mathbb{S}}}^{N} : D m u = K ( ∣ y ∣ ) u m ∗ − 1 , u > 0 in S N , u ∈ H m ( S N ) , {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}), where K ( ∣ y ∣ ) K\left(| y| ) is a positive function, m ∗ = 2 N N − 2 m {m}^{\ast }=\frac{2N}{N-2m} is the Sobolev embedding critical exponent, N > 2 m + 2 N\gt 2m+2 . D m {D}^{m} is the 2 m 2m order differential operator given by D m = ∏ l = 1 m − Δ g + 1 4 ( N − 2 l ) ( N + 2 l − 2 ) , {D}^{m}=\mathop{\prod }\limits_{l=1}^{m}\left(-{\Delta }_{g}+\frac{1}{4}\left(N-2l)\left(N+2l-2)\right), where Δ g {\Delta }_{g} is the Laplace-Beltrami operator on S N {{\mathbb{S}}}^{N} , S N {{\mathbb{S}}}^{N} is the unit sphere with Riemann metric g g . 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.