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

Abstract

In this article, we are concerned with the following prescribed curvature problem involving polyharmonic operator on SN{{\mathbb{S}}}^{N}: Dmu=K(∣y∣)um∗−1,u>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}), where K(∣y∣)K\left(| y| ) is a positive function, m∗=2NN−2m{m}^{\ast }=\frac{2N}{N-2m} is the Sobolev embedding critical exponent, N>2m+2N\gt 2m+2. Dm{D}^{m} is the 2m2m order differential operator given by Dm=∏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), where Δg{\Delta }_{g} is the Laplace-Beltrami operator on SN{{\mathbb{S}}}^{N}, SN{{\mathbb{S}}}^{N} is the unit sphere with Riemann metric gg. 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.

Keywords

• polyharmonic operator
• non-degeneracy
• pohozaev identities
• 35b33
• 35j91