In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{ equation*} -{ \Delta u}\sts{ x} -{ \bm\alpha}\sts{ N,\lambda} \int_{ \R^N} { \frac{ u^{ p}\sts{ y}}{ \pabs{ \,x-y\,}{ \lambda}} }\diff{ y}\, u^{ p-1}\sts{ x} =0,\quad x\in \R^N \end{ equation*} where $N\geq 3$, $0<\lambda<N$, $p=\frac{ 2N-\lambda}{ N-2}$ and ${ \bm\alpha}\sts{ N,\lambda}$ is a normalized constant such that $ u(x)=\left(1+|x|^2\right)^{ -\frac{ N-2}{ 2} }$ is a bubble solution of the equation \eqref{ NLH}. It solves an open nondegeneracy problem in \cite{ MWX:Hartree, GMYZ2022cvpde} and generalizes the partial nondegeneracy results in \cite{ DY2019dcds, GWY2020na, LTX2021} to the full range $0<\lambda<N$. The key observation is that by use of the stereographic projection $\mathcal{ S}$, the weighted pushforward map $\mathcal{ S}_*$ is one-to-one map between the null space of the linearized operator and the spherical harmonic function subspace $\mathcal{ H}_1^{ N+1}$ of degree one.
Categories: math.AP